Boil the pasta: Bring a large pot of salted water to a boil.
Cook pasta until just al dente. Reserve 1 cup of pasta water, then drain.
Pat shrimp dry: While the pasta cooks, pat shrimp dry with paper towels. Season with salt, pepper, and a pinch of red pepper flakes.
Sear the shrimp: Heat 1 tbsp olive oil in a large skillet over medium-high.
Add shrimp in a single layer. Cook 1–2 minutes per side until just pink and opaque. Remove to a plate.
Sauté aromatics: Lower heat to medium.
Add remaining 1 tbsp olive oil. Add garlic (and more red pepper flakes if you like). Cook 30–60 seconds until fragrant, not brown.
Wilt the spinach: Add spinach to the skillet with a pinch of salt.
Toss until wilted (2–3 minutes). If using frozen spinach, add and warm through, breaking up clumps.
Build the sauce: Pour in broth and bring to a gentle simmer. In a small bowl, whisk Greek yogurt with a few tablespoons of hot pasta water to loosen it.
Off the heat, stir the yogurt into the skillet. Add Parmesan and lemon zest.
Adjust consistency: Return skillet to low heat. Add drained pasta and a splash of reserved pasta water, tossing until glossy and lightly creamy.
Add more water as needed to coat.
Finish with shrimp: Add the cooked shrimp and any juices from the plate. Warm gently for 1 minute. Stir in lemon juice.
Taste and season with salt and pepper.
Optional richness: Swirl in 1 tbsp butter for a silkier finish and add chopped parsley if using.
Serve: Plate with extra Parmesan, black pepper, and a squeeze of lemon. Enjoy right away.