Preheat oven to 400 degrees F.
Slice potatoes (about ¼ inch thick). Add a single layer of potatoes to a quiche dish, pie plate or a 10" cast iron skillet. Cover sides of dish and bottom with potatoes, but don't overlap.
Once the dish is covered with the single layer of potatoes, remove the sliced potatoes and place in a bowl. Add 3 Tablespoons olive oil, ¼ tsp. salt and ¼ tsp. pepper. Mix well and re-cover your quiche dish, pie plate or skillet with a single layer of oiled potatoes. Pour on any extra oil. Bake in the oven for 45-50 minutes until roasted.
In the same bowl you used for the potatoes, mix small broccoli florets and onions with 2 Tablespoons olive oil and a sprinkle of salt. Spread broccoli out on a cookie tray and bake in oven for 20 minutes.
Add eggs, milk, cheese, ½ tsp. salt, ½ tsp. pepper, and ½ tsp. thyme and garlic to a bowl. Add broccoli and onion mixture to eggs, once roasted. Whip mixture well with a fork.
Once potatoes are roasted, remove from oven. Turn oven down to 350 F. Add egg mixture to dish with roasted potatoes. Put back in oven and bake for 30-35 minutes or until eggs are set. Remove from oven and serve.