Prepare broccoli and set aside.
Heat the olive oil in a large nonstick skillet and sauté the onion and bell pepper over medium high heat for about 6 minutes, stirring occasionally, until soft and onion is starting to turn golden. Add garlic and cook one more minute. Transfer onion mixture to bowl and allow to cool off.
Spray a ceramic or glass baking dish with at least 2-inch sides (I used a 10-inch deep pie dish) with cooking spray. Arrange the bread cubes in the bottom of the casserole. Place broccoli florets and pieces over and among bread cubes, distributing evenly. Spoon onion mixture around broccoli, then follow with cheeses, sun-dried tomatoes, and thyme.
In a medium bowl (with spout if possible), beat eggs, milk, mustard, salt, pepper, and Tabasco. Pour mixture around edge of baking dish so that eggs seep in without disturbing arrangement of vegetables and cheese.
Cover baking dish with plastic wrap and refrigerate overnight or at least 8 hours.
Preheat oven to 350 degrees F. Remove the plastic wrap from dish and bake strata for 45 minutes. The top of the strata should be golden brown. You can test for doneness by inserting a knife into the middle; it should come out clean.
Allow to rest 15 minutes at room temperature before serving.