Preheat oven to 350 degrees F (175 degrees C). Spray a baking sheet lightly with cooking spray.
Season chicken cutlets with salt and pepper.
Spray a large skillet lightly with cooking spray. Heat over medium-high heat. Saute chicken cutlets until lightly browned and the juices run clear, 8 to 12 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Transfer to a deep container; pull chicken apart with 2 forks.
Heat olive oil in the hot skillet. Saute green bell peppers, yellow bell pepper, red bell pepper, portobello mushroom, and red onion until tender, about 7 minutes. Transfer vegetables to the container with the chicken. Add Monterey Jack cheese, mild Cheddar cheese, sharp Cheddar cheese, cumin, paprika, chili powder, and garlic. Mix well; season filling with salt and pepper.
Whisk flour and eggs together in a bowl. Add milk and water, stirring to combine. Add butter and 1/4 teaspoon salt; beat until batter is smooth.
Heat a lightly oiled griddle over medium high heat. Pour about 1/4 cup batter onto the griddle for each crepe. Tilt the pan with a circular motion so that the batter coats the surface evenly. Cook crepe until bottom is light brown, about 2 minutes. Loosen with a spatula; turn and cook the other side until lightly browned, about 30 seconds more. Place crepe on a plate. Repeat with the remaining batter.
Spoon chicken filling into a crepe, wrap, and place on the prepared baking sheet. Repeat with remaining crepes and filling
Bake in the preheated oven until cheeses melt, 5 to 10 minutes.