Preheat oven to 375ºF. Trim excess fat from chicken thighs, but don’t go crazy—fat equals flavor. Season thighs generously with salt and pepper on both sides. Heat olive oil in a large, lidded, ovenproof sauté pan over medium-high heat. Add the thighs, skin side down, and cook until the skin is nicely browned, 6 to 7 minutes. Turn thighs and brown the underside for about 5 minutes, reducing the heat to medium halfway through. Transfer thighs to a plate and reserve.
Add shallot to the pan and cook, stirring frequently for about 3 minutes. The shallot should be softening, but not browning—reduce heat, if needed. Add garlic to the pan and cook, stirring, for about 30 seconds. Sprinkle flour in the pan and cook, stirring constantly, for about 2 minutes. You want to cook off the raw flour flavor. Later, the flour will thicken the sauce as everything cooks together.
Add broth and lemon juice, and stir, scraping up any browned bits. Stir in basil and return chicken to the pan, skin side up.
Cover the pan and transfer it to the oven. Roast for 15 minutes. Uncover the pan and continue to roast for another 10 to 15 minutes (10 minutes worked for us). Using an instant read thermometer, check chicken for doneness—it should be at least 165ºF. If not, return it to the oven for about 5 minutes. The sauce should be nicely thickened. If not, transfer chicken to a platter and tent with foil to keep it warm, then heat the sauce on the stovetop. (Alternatively, if it’s too thick, thin with just a little water.) Taste sauce and adjust seasoning with salt and pepper, if needed. To serve, plate chicken thighs and spoon some sauce over it.