Spread 1/4 cup flour on a large plate. Season the chicken with salt and pepper, then dredge in the flour and shake off the excess. Heat a large nonstick skillet over medium-high heat. Add the olive oil and heat until shimmering, then add the chicken and cook until golden and almost cooked through, 5 to 6 minutes per side.
Push the chicken to the edges of the skillet, add the shallot and basil stems to the middle and cook until softened but not browned, about 1 minute. Stir in the remaining 1 tablespoon flour and cook 1 minute. Add the chicken broth and lemon juice and bring to a boil, scraping up any browned bits. Reduce the heat and simmer, turning the chicken occasionally, until cooked through and the sauce is thick, 3 to 4 minutes. Turn off the heat and swirl in 2 tablespoons butter until melted. Add the lemon zest and season with salt and pepper.
Melt the remaining 1 tablespoon butter in a large nonstick skillet over medium-high heat. Add the zucchini noodles and season with salt and pepper; cook, stirring occasionally, until just softened, 2 to 3 minutes. Toss in half the torn basil. Divide among plates and top with the chicken, lemon sauce, remaining basil and red pepper flakes.
Photograph by Ryan Dausch