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 pieces; cover and refrigerate for 2 hours, or preferably, overnight.

Heat about 1/2-inch of oil over medium-high heat in a large frying pan or Dutch oven that can be covered, to 350 to 375 degrees on a deep-fry thermometer. (A 13-inch straight-sided frying pan is perfect.)

Meanwhile, combine the flour, salt, and 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. 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. Line a baking sheet with a double thickness of paper towels and drain the chicken on the towels.

Serve immediately, at room temperature, or chilled.

*Pan Gravy Instructions: Over medium heat, melt butter with the pan drippings in the 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; cook until thickened and bubbly, about 5 minutes. Add salt and pepper, and taste for seasoning.