Arrange a rack in the middle of the oven and heat to 375°F. Coat a 9x13-inch baking dish with olive oil; set aside.
Heat the 1 tablespoon of oil in a large frying pan over medium-high heat until shimmering. Add the mushrooms and carrots and cook, stirring occasionally, until tender, about 5 minutes.
Add the spinach and toss until wilted and almost all of the liquid is evaporated, about 5 minutes. Remove from the heat and stir in the potatoes and red peppers.
Transfer to the baking dish and spread into an even layer. Sprinkle evenly with the cheese.
Whisk the eggs, milk, salt, and pepper together in a large bowl. Pour over the vegetables.
Bake until the top is slightly golden-brown and a knife inserted in the middle comes out clean, 50 to 55 minutes. Let cool for 5 minutes before slicing and serving.