Thanks everyone for participating!
Special kudos to Alex Yi for his discussion, and Alex is correct that this is a take on a classic problem in Game Theory about dividing something fairly when people have differing preferences. Game Theory is a way to use math to help understand people's decision making and is famous for its many applications in economics.
The frosting and sprinkles were just examples here, you can think of even more scenarios, such as a piece of cake that is a mix of chocolate and vanilla or even a piece of cake that is taller in some places than in others. The problem (and solution) can also apply to other scenarios, such as a pizza that has different toppings in different places or even dividing chores for the week.
In understanding the solution it can also be helpful to think of the problem with both Michael and Michelle acting greedy. In other words, both would like to have as much cake as possible! Note that Alex's solution still works in this scenario: If Michael is the one cutting the cake, it is in his best interest to cut it "fairly", because if he cuts it unfairly he is risking Michelle picking the "better" piece.
Those interested can try to extend the solution to three or more people. It gets a little more complicated, but the idea is the same!