Prep the potatoes: Rinse and pat dry. Cut into even pieces so they cook at the same rate.
Toss with 1 tablespoon olive oil, 1/2 teaspoon salt, and 1/4 teaspoon black pepper.
Start the potatoes: Heat a large skillet over medium-high. Add 1 tablespoon olive oil and 1 tablespoon butter. When the butter foams, add the potatoes cut side down.
Cook undisturbed for 4–5 minutes to brown, then toss and continue cooking, stirring occasionally, for 8–10 more minutes until tender and crisp at the edges. Transfer to a plate and keep warm.
Season the shrimp: Pat shrimp dry. Toss with 1/2 teaspoon salt, 1/4 teaspoon pepper, paprika, and dried thyme.
Dry shrimp sear better and won’t steam.
Sear the shrimp: In the same skillet, melt 1 tablespoon butter over medium-high heat. Add shrimp in a single layer. Cook 1–2 minutes per side until just opaque and pink.
Do not overcook. Transfer shrimp to the plate with potatoes.
Build the garlic butter sauce: Lower heat to medium. Add remaining 2 tablespoons butter to the skillet.
Stir in minced garlic and red pepper flakes. Cook 30–45 seconds until fragrant, not browned.
Deglaze and brighten: Pour in the chicken broth and scrape up any browned bits. Simmer for 1–2 minutes to reduce slightly.
Add lemon zest and juice. Taste and adjust salt and pepper.
Bring it all together: Return potatoes and shrimp to the pan. Toss gently to coat everything in the sauce.
Warm through for 1 minute. Sprinkle with chopped parsley.
Serve: Plate immediately with extra lemon wedges. If you love saucy plates, stir in the optional extra tablespoon of butter off the heat for a silkier finish.