Mash garlic and salt into a paste in a small bowl, using the back of a spoon. Stir in pepper and thyme.
With a sharp knife or kitchen shears, remove any excess fat from chicken. Dry the inside with a paper towel. With your fingers, loosen the skin over the breasts and thighs to make pockets, being careful not to tear the skin. Rub the garlic mixture over the breast and thigh meat.
Heat oil and 1 teaspoon butter in a 12-inch cast-iron skillet over medium heat. Add the chicken and cook, turning often, until nicely browned on all sides, about 10 minutes.
Transfer the pan to the oven and roast the chicken until the internal temperature in the thickest part of the thigh reaches 165°F, 50 minutes to 1 hour. Transfer the chicken to a clean cutting board; tent with foil.
Meanwhile, mash the remaining 1 teaspoon butter and flour in a small bowl until a paste forms. Place the pan (use caution, the handle will be hot) over medium-high heat. Add broth and bring to a simmer, stirring to scrape up any browned bits. Gradually whisk in the butter-flour paste a few bits at a time, until the gravy thickens, about 8 minutes. Remove from the heat and let stand for 5 minutes, allowing any fat to rise to the top. Skim off the fat with a spoon. Carve the chicken and serve with the gravy. Garnish with parsley, if desired.