Preheat the oven to 425° F, and lightly grease or line a large, rimmed baking sheet with parchment paper.
Crack the eggs into a medium bowl and whisk together with the Parmesan, 1 teaspoon kosher salt and black pepper. Set aside.
Cut the broccoli into bite-size florets. (I tend to go heavy on the broccoli and often dice the stems and include them, too.) Toss the broccoli and the onion slices with 2 tablespoons of the olive oil, and sprinkle with salt and pepper to taste. (I usually add about 2 pinches of each. As for tossing, you can do that in a large bowl or directly on the baking sheet, as preferred.)
Transfer the veggies to the prepared baking sheet, spreading into an even layer. Roast for 15-20 minutes or until the stems are just tender and the tips of the broccoli are golden brown. Remove from the oven, set aside, and reduce the oven temperature to 300° F.
Heat the remaining tablespoon of oil in an oven-safe 10- to 11-inch sauté pan or skillet set over medium heat, and add the garlic and chile. Cook for a minute or until aromatic, watching closely so as not to burn the garlic. Add the roasted veggies to the pan and mix well. Add the egg and cheese mixture, using a fork or spatula to evenly distribute the vegetables. (It’s ok if the some of the broccoli and onions stick out.) Increase the heat to high and cook for 30 seconds.
Transfer the pan to the 300° F oven and cook until the eggs are just set, about 25 minutes, give or take a few. (Start checking after 20 minutes.) Remove from the oven sprinkle with additional Parmesan cheese, if desired.