Season each quail inside and out with salt and pepper.
Transfer to 1 to 2 large bowls, add oil, thyme, and garlic, and toss to combine. Refrigerate, covered, at least 1 hour and up to 48 hours.
Preheat oven to 500°F.
Remove 1 quail from marinade. Stuff cavity with 1 ounce goat cheese and 1 sprig rosemary and tie legs together loosely with kitchen string. Wrap 1 strip bacon around breast and transfer quail, breast side up, to rimmed baking sheet. Repeat with remaining quail, using 2 to 3 baking sheets. Roast until just cooked through (cut into inner thigh; meat will still be slightly pink), about 15 minutes.
If you prefer, the quail can be grilled rather than roasted. To avoid flare-ups from dripping fat, cook the bacon separately and stuff it in the cavity of each bird along with the cheese.
