Pat the chicken dry with paper towels. In a large bowl, combine 1 tablespoon of the flour, 1/2 teaspoon salt, and 1/2 teaspoon black pepper. Add the chicken and turn to coat.
Heat the oil in a large skillet over medium-high heat. Add the chicken and cook until golden brown on all sides (and a nice crust forms). Work in batches if necessary to prevent crowding the pan, then return all the chicken to the pan.
Add 2 tablespoons of the butter and garlic and stir to coat the chicken. Cook until the butter melts, the garlic is toasty, and the chicken is cooked through, stirring frequently.
Transfer the chicken to a shallow serving bowl and cover with foil to keep warm.
Melt the remaining tablespoon of butter in the same pan over medium-high heat. When the butter is bubbly, whisk in the remaining tablespoon of flour until blended.
Whisk in the broth plus the juice of 3 to 4 of the lemon wedges and bring to a simmer. Simmer for 2 minutes, until the sauce thickens, scraping up any browned bits from the bottom of the pan.
Season to taste with salt and black pepper. Spoon the sauce over the chicken and garnish with parsley. Serve with the remaining lemon wedges on the side.