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 5 minutes. Remove 1/2 of the mixture and set aside in a bowl.

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 with all 4 sides of the egg, cooking until the bottom is set, about 2 minutes. Flip over and cook until bottom is slightly browned, 1 to 2 minutes more.

Slide omelet onto a tortilla, then slide tortilla back onto the pan. Place 1 cheese slice onto the omelet and top with another tortilla. Cook until the bottom is browned, about 2 minutes. Flip and brown opposite side, about 1 minute more. Slide quesadilla onto a plate and cut into 4 pieces.

Beat remaining 2 eggs and 1 1/2 tablespoons milk together in a bowl. Grease the same pan with remaining 1 teaspoon butter. Repeat steps 3 and 4 with reserved vegetable mixture and remaining cheese and tortillas.