Boil salted water and cook linguine until al dente, 8–9 minutes. Reserve 60ml pasta water before draining. Do not rinse.
Pat shrimp dry. Season with salt, pepper, and red pepper flakes.
Heat 1 tbsp butter and olive oil in a large skillet over medium-high. Sear shrimp 1.5–2 minutes per side until pink and curled to a loose C shape (63°C / 145°F internal). Remove to a plate.
Reduce heat to medium. Add remaining 2 tbsp butter. Cook garlic 60–90 seconds, stirring constantly, until fragrant.
Pour in chicken broth. Scrape the pan. Add lemon juice and zest. Simmer 2–3 minutes until reduced by a third.
Add drained linguine and reserved pasta water. Toss over medium heat 1–2 minutes until sauce is glossy and clings to the pasta.
Return shrimp to the pan. Toss 30–45 seconds to warm through. Remove from heat. Top with parsley and serve immediately.