In a medium sauté pan combine the oil, thyme, and red pepper flakes. Set the pan over medium-high heat to warm the oil mixture for 1 to 2 minutes. Drain, rinse, and coarsely chop the artichokes and add to the pan along with the garlic, salt, and pepper. Cook for 3 to 4 minutes, stirring occasionally. Remove from the heat. Add the goat cheese, sun-dried tomatoes, and basil. Mix to evenly distribute the ingredients. Set aside to cool.
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Place each chicken breast between two sheets of plastic wrap and, with a meat mallet or the back of small pan, pound to flatten to an even thickness of about ¼-inch. Place the breasts, skin side down, and spread each one with a quarter of the stuffing. Fold the breasts in half over the stuffing and use toothpicks to skewer the sides closed. Brush both sides with oil and season with salt and pepper.
Brush the cooking grates clean. Grill the chicken breasts over direct medium heat, with the lid closed as much as possible, until the juices run clear and the cheese is melted, 8 to 12 minutes, turning once (if flare-ups occur, finish grilling the breasts over indirect medium heat). Remove from the grill and carefully remove the toothpicks. Serve warm.