How do you clean a quesadilla maker?

So, I have this electric quesadilla maker. It’s kind of like a waffle iron, but flatter and cuts the quesadillas into sixths. There are lots of little grooves and crannies in it because it puts shapes (a cactus, maracas, etc.) on the quesadilla. The little grooves/shapes are what makes it hard to clean. No matter what I do, I can’t get the little bits of food/cheese out of the grooves. I’ve tried several different types of soap/sprays, many sponges, and paper towels. It cannot go in a dishwasher because it has a cord.

Does anyone have a similar appliance? Has anyone found a foolproof way to clean it? THANKS!