Bring a large pot of salted water to a boil. Cook the pasta until just al dente, about 3 minutes shy of the package instructions. Reserve a cup of the pasta water before draining. Set aside the pasta while you move on to the chicken.
Pat the chicken dry with paper towels. Toss it in a bowl with kosher salt and all-purpose flour to coat evenly. Heat olive oil with one tablespoon of butter in a large skillet over medium heat. Add the chicken and cook until it’s golden brown and cooked through, about 5-6 minutes. Transfer the chicken to a plate and set aside.
Lower the heat slightly and add one tablespoon of butter to the skillet. Toss in the chopped shallots and cook until softened, about 3 minutes. Stir in the garlic and cook for another 30 seconds until it’s fragrant but not browned.
Pour in the white wine (or chicken broth) to deglaze the pan. Scrape the bottom with a wooden spoon to release all the flavorful brown bits. Simmer the mixture until reduced by half, which will take about 3 minutes.
Reduce the heat to low and add ½ cup of reserved pasta water, lemon zest, lemon juice, and red pepper flakes to the skillet. Stir well before returning the chicken to the pan. Add the cooked pasta and the remaining butter, tossing everything to coat evenly in the sauce.
Add the grated Parmesan and fresh parsley to the skillet, tossing gently until combined. Turn off the heat and serve immediately, garnished with extra Parmesan and parsley if desired.