Rinse the chicken under cold, running water. Pat dry with paper towels.
In a large bowl, combine the buttermilk, 1 tablespoon salt, 1 teaspoon pepper, the garlic, and thyme. Add the chicken, cover and refrigerate for at least a couple of hours, or preferably, overnight.
Heat about 1/2 inch oil over medium-high heat in a large frying pan or Dutch oven something that can be covered to 350 degrees to 375 degrees F on a deep-fry thermometer (a 13-inch straight sided frying pan is perfect).
Meanwhile, combine the flour, salt, and teaspoon pepper in a heavy re-sealable plastic bag. Working with 2 or 3 pieces at a time, take the chicken out of the buttermilk, put it in the bag, close the bag and shake well to coat the chicken with the flour. Put the chicken on a baking sheet. Continue until all of the chicken has been coated.
When the oil is hot, carefully place the chicken into the pan with tongs, skin side down. You may have to cram the pieces in dont worry about it; you can rearrange later. The important thing is to get them in before hot oil splatters all over you. Cover, and cook 5 minutes. Then uncover the pan and cook until the skin is nicely browned, 5 more minutes.
Turn the chicken and continue to cook until well browned all over and cooked through, 6 to 8 more minutes for the breast meat, 8 to 10 more minutes for the dark meat. If the chicken starts to brown too quickly, turn the heat down a little (you probably wont have to).
Line the baking sheet with a double layer of paper towels and drain the chicken on the towels. Serve immediately, at room temperature, or cold. Pan Gravy: Over medium heat, melt butter with pan drippings in skillet, add flour, and whisk constantly until bubbly and slightly golden-brown, 3 to 5 minutes, scraping up brown bits from the bottom of the pan. Gradually add milk, whisking constantly, add thyme and cayenne, and cook until smooth and thickened, about 5 minutes. Add salt and pepper and taste for seasoning.