Preheat oven to 400 degrees. In a large skillet, heat oil over medium-high. Add spinach and cook, stirring constantly, until wilted, about 2 minutes. Transfer to a colander and drain, pressing with a rubber spatula to release as much liquid as possible. Divide bread and spinach among four small, shallow 8-ounce baking dishes.
In a medium bowl, whisk together eggs, milk, and 3/4 cup cheese. Season with 1/2 teaspoon salt and 1/4 teaspoon pepper. Divide mixture evenly among baking dishes and top with 1/2 cup cheese. Set dishes on a rimmed baking sheet and bake until set in middle and golden brown on top, 20 to 25 minutes.