Preheat oven to 375 degrees and line a baking sheet with foil or use a nonstick baking sheet. Remove butter from refrigerator and allow to come to room temperature. Slice chile in half, remove seeds and membrane (spiny white innards) if averse to heat and cut into a fine dice. Peel and mince shallot. Mince garlic. Halve lime. Cut broccoli crown in half. Coarsely chop parsley. Rinse tilapia filets and pat dry.

In a medium mixing bowl, add room-temperature butter, half the chile, half the shallot, half the garlic, parsley (reserving some chili and parsley for garnish), ½ tsp. salt, and juice of the lime to taste. Incorporate the ingredients together with a spoon or spatula. Mold into 2 equally-sized circular portions and place in the refrigerator to chill and firm up.

Place flour in a shallow dish or on a plate. Season each side of fish with a pinch of salt and pepper. Dredge fish in flour until completely coated.

Place broccoli halves on the prepared baking sheet. Drizzle halves with 1 tsp. olive oil and season with salt and pepper to taste. Roast in oven for 7-10 minutes, or until tender and edges are beginning to brown. Remove from oven and keep warm until ready to serve.

In a medium oven-safe pan, add 2 tsp. olive oil, remaining shallot, remaining chili, and remaining garlic over medium-high heat. Add fish and cook 2-3 minutes per side, or until golden brown. Top each fillet with a portion of butter and place pan in oven on top rack. Bake 2 minutes, or until butter is melted. Remove from oven and keep warm until ready to serve. Salt and pepper to taste.

Place a roasted broccoli half in the middle of the plate. Lay a piece of fish against the broccoli. Spoon melted compound butter from pan over the fish. Garnish with remaining parsley, any remaining lime slices, and a crack of fresh black pepper, if desired.