Spray an 8-inch casserole dish and set aside.

Place broccoli in a microwave safe bowl, add 2 Tbs water, cover with plastic wrap and microwave for 2 minutes or until tender.

Heat oil in a sauté pan on medium-high heat, and add onions and a pinch or two of salt. Saute onions until golden, about 7-10 minutes. Drain broccoli and add to the pan and stir well with onions. Remove from heat.

In a medium bowl, whisk together eggs, milk, any remaining salt, and pepper.

Lay the remaining bread on top, and pour egg mixture over all, pressing down with a spatula so that eggs and milk are absorbed. Top with remaining cheese.

Let the casserole sit for 10-15 minutes, and preheat the oven to 350 degrees.

Bake strata, covered with foil, for 30 minutes, and then remove foil and bake for another 15-20 minutes or until cheese is golden.

Let casserole sit for a few minutes before serving.