Epistemic status: babble all the way down, not pruning. But I believe my approach is better than most of other answers here.

The error from other LWers is not separating the evaluation of lessons to the evaluation of tests.

Students should be allowed to give good teachers a bonus. For each lesson, in any moment of the lesson, the students should have the possibility to rate the teacher's performance on some metrics. Think on a mobile application that does that. Do you know when you take a ride with an Uber and immediately after finishing the ride you rate it? We should have the same possibility of rating teachers after their lessons (up until some limit, e.g., you had your lesson on Monday, you won't be able to rate it on the next month, you have until a week to rate this lesson). The teacher should be paid a bonus when he gets good scores. This bonus would be added lesson by lesson to teacher's account.

Let's say each week I have three different lessons with professors A, B and C.
Professor A gives me 2 hours lesson/week.
Professor B gives me 4 hours lesson/week.
Professor C gives me 6 hours lesson/week.

For each two hours lesson, the student gains one point to spend. So, I have 6 points to spend on spend on professors A, B, C in any way I choose to.

The professor A, I've just watched his lesson and I loved it. I give him 3 points. Professor B is good too, I like him, but I will just give him two points. Professor C is not that good teacher, but he seems to be working hard on these particular difficult topics, I'll give him one point.

On the end of the month, good teachers will be rewarded by how good their performance were on THE LESSONS. I haven't spent time thinking on a good function to convert the scores received by the teacher on this week to money, but it doesn't seem hard to create a fair one.

We should have a separate rating system for the evaluation of the tests applied by the teacher, so we could separate the feelings that appear on our heart when we compare the quality of the lessons to the difficulty of the problems posed by the professor on his test. We know when we go bad on an exam, "that's the teacher's fault". So this separate system would be more strict, asking several questions like "How difficult was this test? How many hours have you studied before doing it? The questions on the test were related to things taught on the lessons? How do you compare the difficult of the test questions to the difficult of the lessons' questions? Leave a comment about the test on the following Entry box". Obviously I haven't pruned these questions, they just arrived at my mind, but certainly there exist a very good set of questions that could let us investigate how well the teachers perform in creating tests and also reward them when we detect it.

Thinking again about the first system, it should also have some questions about the lesson. "How good the professor explains the concepts? How organized he is? Did you learn the concepts? How do you rate the difficulty of the topics this teacher is trying to explain to you? [Leave here what kind of questions you believe would improve this questionnaire]. Leave a comment about the lesson on the following Entry box".

It shouldn't be needed for a student to answer these questions to give all his points for a teacher. But we could weight the student points by how many questions he answered. For example, if I gave you 3 points and I said why I've done this, this weighs more than a student that gives you 3 points but doesn't explain why he does that. Justified rating is worth more than unjustified rating.

this advantages teachers with larger classes.

Your reward function can take in consideration the number of students that participated on the lesson, the number of students the rated the professor, and also you could average the scores, I don't know, come on, you can create a function that is fair for any class size, you just have to think about what function you will use

Where does the money come from?

Diminish all salaries in x%. Now you can redistribute this money more fairly, proportional to performance.

My second-favorite teacher in undergrad was relatively unpopular because he taught very difficult classes, at least some of which were required to graduate.

That's why it is important to evaluate the LESSONS every week. And when the test comes, this is a different evaluation. This professor was unpopular due to difficult tests, not to bad lessons, right?

Most universitites have already systems where students evaluate their teachers at the end of the year and the scores do figure into administrative decisions of the university

That's the problem. At the end of the year you are evaluating the "teacher", which means

If I find the teacher a good professor and I give him +5 points, but I sucked at his tests, and I give -10 points for my bad feelings for doing bad on the test, the final evaluation of the teacher is "tis teacher is bad" == -5

If the system rates week by week, we could detect misuse of the system if we suddenly see bad lessons ratings close to the test application (right after the test, for example).

I don't think this is how market wages work. If it is known that the average teacher gets a $100 bonus, the school will offer $100 less in base pay than it would otherwise.
Maybe not right now, when the change is introduced. But in the following years, the wages will raise slower than they would otherwise, until the balance is achieved.

It doesn't seem bad to pay a little less for the average teacher with average lessons, and pay a little more for the above average teacher with above average lessons. It seems like arbitrage. You do good lessons, you earn more. Why not? And if you can now detect which teachers are much worse than average, you can fire them and get even more students that are interested in this school full of good teachers, because the bad ones can't stay