Combine the salsa ingredients, including ½ teaspoon of the salt, and mix well. Reserve ¼ cup of the salsa to use as a stuffing for the chicken breasts. (The salsa is best if made a few hours before serving. Keep it covered and refrigerated.)
Prepare the grill for direct cooking over medium heat (350° to 450°F).
Place the chicken on a work surface, skin (or smooth) side down, and then cut a large pocket in the thickest part of each breast. Push about 1 tablespoon of the cheese inside the pocket, and then spread about 1 tablespoon of the reserved cranberry mixture on top of the cheese. Press the pocket closed. Brush both sides of the chicken with oil and season evenly with the remaining ½ teaspoon salt, the pepper, and thyme.
Grill the chicken, skin (or smooth) side down first, over direct medium heat, with the lid closed, until the meat is firm to the touch and opaque all the way to the center, 8 to 12 minutes, carefully turning once. Remove from the grill and let rest for 3 to 5 minutes. Serve warm with the remaining cranberry salsa.