Preheat oven to 350 degrees. Lightly butter a 9x13-inch baking dish; set aside.
Steam broccoli for a few minutes (do not fully cook the broccoli). Place the broccoli in a large mixing bowl; set aside.
In a medium saucepan set over medium heat, add the olive oil. Add the mushrooms and brown, about 6-7 minutes, stirring occasionally. Add the onion and garlic and cook until translucent and fragrant. Transfer to the bowl with the broccoli.
In the same saucepan, melt 2 tablespoons of the butter. Whisk in the flour and cook for 1 minute.
Slowly whisk in the milk and half-n-half until smooth. Bring the mixture to a low simmer, whisk until thick. Remove from the heat and whisk in the salt, pepper, paprika, cayenne pepper and Greek yogurt.
Pour sauce into the large mixing bowl (with the veggies). Add the egg, 1 cup of the cheddar cheese, and ½ cup of the breadcrumbs, mix until combined. Pour into the prepared baking dish.
In a medium microwave safe mixing bowl, melt the remaining tablespoon of butter. Toss the remaining breadcrumbs in and stir to combine, then mix in the remaining cheese. Sprinkle evenly over the broccoli mixture.
Place in the oven and bake for 30-40 minutes or until it bubbles and browns (place a piece of foil over top to prevent over-browning at the 30 minute mark). Remove from the oven and let stand for 10 minutes before serving.
Notes
-Try adding chicken or ham to this dish for a complete meal!