In an oven-safe skillet (I use my 10-inch cast iron skillet), over medium high heat, cook the bacon until it begins to crisp (about 5 minutes). Add the oil, onions, and mushrooms, and saute for 4 minutes, until softened. Add the broccoli and saute for another 4 minutes, until it begins to soften.
Turn off the heat, removing the skillet from the hot surface. Sprinkle the cubed bread evenly over the veggies. Once the skillet cools, sprinkle the cheese evenly over the bread. While you're waiting for the skillet to cool, I suggest whisking the egg mixture...
In a medium-size bowl, whisk together the eggs, milk, dijon mustard, chopped rosemary, salt, and pepper. Once the skillet is cool, pour the egg mixture over the cheese, cubed bread, and veggies. The cubed bread won't be fully submerged in the egg mixture.
At this point, you can bake the strata, but for the best flavor and texture I recommend covering the skillet with foil (or a cover) and placing the skillet in the fridge overnight.
In the morning, remove the skillet from the fridge, along with the covering. Preheat the oven to 350F. Once the oven is ready, bake the strata, uncovered, for 25-30 minutes until the egg mixture is set. If you'd like a crispy/toasty top, turn off the oven and broil the strata on low for just a couple of minutes.
Allow the strata to rest for about 10 minutes to set, and then serve.
Notes
*If you're dairy-free, coconut or almond milk may work in this recipe. Both options will slightly change the flavor of the strata. **I grow a ton of rosemary in our garden, so this is my go-to herb for so many dishes. This strata would also be delicious with fresh dill or chives in place of the rosemary. Use what you love and have on hand.