Preheat oven to 350°F.

In a large, ovenproof skillet over medium heat, cook vegetables in one tablespoon of oil until tender. Season with salt and pepper to taste. Transfer to a bowl and set aside.

With skillet still over medium heat, add remaining oil, and place a single layer of bread on the bottom. Fry without moving the bread for about 3 minutes.

Top bread with reserved vegetables, spreading to make an even layer.

In a small bowl, whisk together eggs and milk, pour over bread and vegetables, then top with cheese and basil.

Bake until cheese is melted and slightly browned and eggs are set, 15 to 25 minutes.