Bring a large pot of salted water to a boil. Add the pasta and cook until just al dente. Reserve 1 cup of the pasta cooking liquid, then drain*.

Pat the chicken dry. Place in a bowl and toss with the salt and the flour.

In a large, deep skillet, heat the oil and 1 tablespoon of the butter over medium. Add the chicken and cook on all sides untiled browned and cooked through, about 5 minutes. Remove to a plate.

Add 1 tablespoon butter and the shallot to the skillet. Cook until the shallot is beginning to soften, about 3 minutes. Stir in the garlic and let cook 30 seconds to 1 minute, until fragrant.

Pour in the wine and stir to scrape up the browned bits from the bottom of the skillet. Let simmer until reduced by half, about 3 minutes.

Reduce the heat to low. Add ½ cup of the reserved pasta water, lemon zest, lemon juice, red pepper flakes, chicken, pasta, and remaining 2 tablespoons butter. With tongs, toss the noodles with the chicken so that they are coated with the garlic-lemon sauce. Add the Parmesan and parsley. Continue tossing until silky and combined. Enjoy immediately, with additional Parmesan to taste.