Preheat the oven to 400 degrees F (200 degrees C).
Trim root end from garlic cloves and remove skin. Place cloves on a large square of aluminum foil. Place thyme sprigs among the garlic, sprinkle with salt and pepper, and drizzle with olive oil. Fold up the sides of the foil to create a packet and place on a baking sheet.
Bake in the preheated oven until garlic is soft and golden brown, 45 minutes to 1 hour. Discard thyme and let garlic cool.
Place cream cheese into a medium bowl. Add garlic on top and mash cloves with a fork. Season with additional salt and cracked black pepper to taste, and stir mixture until evenly combined. Alternately, you may use a small food processor for a smoother consistency.