Heat the oil in a large skillet over medium heat. Add onion, thyme, and garlic and cook until soft. Add in the vegetables and ¼ teaspoon of the salt and cook, remove while still somewhat crips. Set aside.
Whisk together eggs, milk, mustard, parsley, pepper, and remaining salt.
Butter a 9×13 baking dish and place the bread across the bottom. Spread veggie mixture over the bread then sprinkle with ½ of the cheese.
Pour the egg mixture over the vegetables and top with remaining cheese. Let stand for about 10–15 minutes to let the bread soak up some of the liquid.
Bake in a 375º oven for about 30 minutes or until firm in the center and browned on top.
Let stand for about 10 minutes before cutting into squares and serve.