Preheat the oven to 375 degrees F. Butter a 9-by-13-inch baking dish.
Heat the oil in a large skillet over medium heat. Add the onions and cook, stirring occasionally, for 3 minutes. Add the garlic, bell peppers, broccoli, cauliflower and 1 teaspoon of the salt. Cook, stirring occasionally, until the onions are soft, about 5 minutes. Remove from the heat and let cool.
Meanwhile, in a large bowl, beat the eggs with the milk, the remaining 1 teaspoon salt and 2/3 cup of the cheese. Add the bread cubes and the vegetables and toss to combine. Transfer to the prepared baking dish, pressing down gently to soak the bread.
Sprinkle the strata with the remaining 1/3 cup cheese. Cover with foil and bake for 30 minutes. Remove the foil and continue to bake until the top is golden brown and the strata is cooked through, about 20 minutes.