Bring a pot of salted water toward a boil while you prep the chicken. Mix 1/4 cup flour with 1/2 teaspoon salt, 1/4 teaspoon pepper, and 1/2 teaspoon garlic powder. Lightly dredge the two pounded chicken breasts and shake off the excess.
Heat 1 tablespoon olive oil and 1 tablespoon butter in a large skillet over medium-high heat. Cook the chicken about 3 to 4 minutes per side, until golden and 165°F. Transfer to a plate.
Reduce the skillet to medium. Add 1 minced garlic clove and cook 30 seconds. Pour in 1/4 cup white wine and scrape up the browned bits; simmer about 1 minute. Add 1/2 cup chicken broth, 2 tablespoons lemon juice, 1/2 teaspoon zest, and optional 1/2 teaspoon Dijon. Simmer 3 to 4 minutes until slightly reduced.
Meanwhile, cook 6 ounces angel hair until just shy of al dente. Reserve at least 1/4 cup pasta water, then drain. Toss the pasta with 1 tablespoon butter, 1 small minced garlic clove, and 1 tablespoon pasta water; add another splash if needed.
Take the sauce off the heat and whisk in 1 tablespoon cold butter until glossy. Stir in 1 tablespoon parsley. Return the chicken to the skillet for 1 minute, spooning sauce over it.
Divide the angel hair between two plates, top with the chicken, and spoon the lemon-butter pan sauce over both.