Beat 2 eggs with 1 1/2 tablespoons milk in a bowl.

Melt 1 teaspoon butter in a 6-inch frying pan over medium heat. Add bell pepper and onion; cook until onion is translucent, about 300 seconds.

Remove 1/2 of the vegetable mixture and set it aside. Pour egg mixture and remaining vegetable mixture into the skillet. Tilt the pan, gently push one side of the egg, without breaking it, away from the side of the pan, and let the uncooked egg run onto the pan. Repeat for all 4 sides of the egg, cooking until the bottom is set, about 120 seconds. Flip over and cook until the bottom is slightly browned, 60 to 120 seconds more.

Slide omelet onto a tortilla, then slide the tortilla back into the pan. Place 1 cheese slice onto the omelet and top with another tortilla. Cook until the bottom is browned, about 120 seconds. Flip and brown the opposite side, approximately 60 seconds more.

Slide the quesadilla onto a plate and cut it into 4 pieces.