In a large skillet, melt 2 Tbsp butter over medium heat. Add the english muffin crumbs and cook, stirring occasionally, until golden, about 5 minutes. Transfer to a plate. Wipe out the skillet and melt remaining 1 Tbsp butter. Add the shallot, 1/2 tsp. salt and 1/4 tsp. pepper and cook until softened, stirring often, about 2 minutes. Add the broccoli and cook for 1 minute. Add 1/2 cup water, increase heat to medium-high and simmer, stirring occasionally until the water is evaporated and the broccoli is crisp tender, 3-5 minutes. Transfer to the prepared baking dish. Sprinkle with 1 cup cheese, pour the egg mixture on top.