Prep the shrimp and broccoli: Pat shrimp dry with paper towels so they sear instead of steam. Toss them with a pinch of salt, pepper, and the paprika.
Cut broccoli into small, even florets for quick, consistent cooking.
Steam-sauté the broccoli: Heat 1 tablespoon olive oil in a large skillet or sauté pan over medium-high. Add broccoli and a pinch of salt. Cook 2 minutes, stirring once.
Pour in the chicken broth, cover, and steam 2–3 minutes until bright green and crisp-tender. Transfer broccoli to a plate.
Sear the shrimp: In the same pan, add 1 tablespoon olive oil and 1 tablespoon butter. When hot and foamy, add shrimp in a single layer.
Cook 1–2 minutes per side until just opaque and lightly golden. Remove shrimp to the plate with the broccoli.
Build the garlic butter sauce: Lower heat to medium. Add remaining 2 tablespoons butter to the pan.
Stir in minced garlic and red pepper flakes. Cook 30–60 seconds until fragrant, not browned. Add lemon zest and a squeeze of lemon juice.
Deglaze and season: Pour in a splash more broth or water if the pan looks dry.
Scrape up any browned bits—they add big flavor. Taste and season with salt and black pepper.
Bring it together: Return shrimp and broccoli to the pan. Toss to coat in the garlicky sauce for 30–60 seconds until heated through.
Add more lemon juice to taste and sprinkle with parsley.
Serve: Spoon over rice or noodles, or serve with crusty bread to soak up the sauce. Finish with an extra crack of pepper and a final squeeze of lemon if you like.