Pat chicken very dry with paper towels. In a bowl, combine smoked paprika, 1 tsp salt, and ½ tsp pepper. Toss chicken to coat evenly. Let sit 10 minutes at room temperature.

In a small pan over medium-low heat, warm 1 tbsp olive oil. Add smoked paprika and stir constantly for 30–45 seconds until fragrant and deepened in color. Transfer immediately to a small dish to cool.

Heat remaining 1 tbsp olive oil in a large heavy-bottomed skillet over medium-high heat. When shimmering, add chicken in a single layer (work in batches if needed). Sear undisturbed 3–4 minutes per side until deeply golden and cooked through (165°F internal temperature). Transfer to a plate and tent loosely with foil.

Reduce heat to medium-low. Add shallots and 2 tbsp butter to the skillet. Cook 3–4 minutes until soft and translucent. Add minced garlic and the cooled paprika mixture. Stir constantly for 60–90 seconds until fragrant — do not let garlic brown.

Pour in white wine, scraping up all browned bits from the bottom of the pan with a wooden spoon. Simmer 2–3 minutes until reduced by half. Whisk in heavy cream and remaining cold butter, adding it piece by piece, until fully melted and sauce thickens slightly — about 90 seconds. Keep heat low; do not allow sauce to boil hard.

Cook pasta in a large pot of well-salted boiling water until 1 minute shy of al dente per package directions. Before draining, reserve ½ cup pasta water.

Return chicken and any accumulated juices to the skillet. Add drained pasta, lemon zest, 2 tbsp parsley, and Parmigiano-Reggiano. Toss gently with tongs over low heat for 1–2 minutes until evenly coated and warmed through. Add reserved pasta water 1 tbsp at a time if sauce is too thick.

Remove from heat. Stir in remaining parsley. Taste and adjust salt, pepper, or add a small squeeze of lemon juice. Serve immediately, garnished with extra Parmigiano, fresh parsley, and a pinch of flaky sea salt.