Combine bopis, beaten eggs, and flour until well combined. Form square patties.
Fry patties in a pan with butter over medium heat for a few minutes.
Take bread and spread butter on one side of the bread slices and lightly toast.
On the non-buttered side of the bread place a small amount of both cheese, then the bopis patty and more cheese. Top with another slice of bread. Repeat for other sandwiches.
Spread mayonnaise on the outer side of the bread slices and cook the grilled cheese until the cheese is melted and the bread is toasted.