Heat oven to 350° F. If the bread is fresh, place it on a baking sheet and transfer to the heating oven until lightly toasted, about 10 minutes.
Heat the oil in a large skillet over medium heat. Add the bell peppers and onion and cook, stirring occasionally, for 10 minutes. Add the garlic, thyme, 1 teaspoon of the salt, and 1/2 teaspoon of the pepper. Cook for 1 minute. Remove from heat; set aside.
Butter a 2-quart casserole and place it on a foil-lined baking sheet. Place the bread in the prepared dish, overlapping the slices slightly. Spread the pepper-onion mixture over the bread. Sprinkle with 1/2 cup of the Gruyère. In a small bowl, lightly beat the eggs, milk, and the remaining salt and pepper. Pour the egg mixture over the layers in the casserole. Sprinkle with the remaining Gruyère. Bake, uncovered, until the strata is puffed and browned, 30 to 40 minutes.
In Advance: Assemble the strata but do not bake it. Cover and refrigerate for up to 24 hours. Bake as directed, adding 5 to 10 minutes to the cooking time.