Preheat the oven to 375F.
Heat the oil in a large saute pan over medium heat. Add the onions and cook for 3 minutes.
Add the garlic, bell peppers, broccoli, cauliflower, and 1 teaspoon of the salt. Cook 5 more minutes, until the onions are soft. Allow to cool.
In a large bowl, beat the eggs, milk, the remaining 1 teaspoon of the salt and 2/3 cup of the cheese.
Grease a 13 x 9 baking dish, add the bread cubes and the cooked veggies, and toss to combine.
Pour the egg mixture into the baking dish, pressing down the bread mixture gently to soak up some of the eggs and sprinkle with the remaining 1/3 cup of cheese.
Cover with foil and bake for 30 minutes.
Remove the foil and continue to bake an additional 20 minutes.