Preheat oven to 350 degrees F (175 degrees C). Spread 1 tablespoon olive oil on a baking sheet.

Heat 1 tablespoon olive oil in a skillet over medium heat; cook and stir shallots until translucent, about 3 minutes. Stir garlic, thyme, salt, and pepper into shallots; cook and stir until fragrant, about 2 minutes. Transfer shallot mixture to a bowl; stir in goat cheese, mozzarella cheese, cream cheese, figs, and basil until well combined.

Pound each chicken breast on a work surface until thin. Spoon about 1/4 cup goat cheese mixture onto the middle of each piece of chicken. Roll chicken around the filling and wrap 2 pieces prosciutto around each chicken roll, covering the opening with 1 layer of prosciutto. Place chicken rolls onto the prepared baking sheet, seam-side down.

Bake in the preheated oven for 20 minutes; turn chicken and continue baking until chicken is no longer pink in the center and prosciutto is crisp, about 20 minutes more. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C). Let chicken stand for 5 minutes before serving.