Let goat cheese (chevre) stand until at room temperature. Preheat oven to 400 degrees F. Line a baking sheet with parchment paper; set aside.
Using a sharp knife, make a lengthwise slit in each whole date. Spread dates open slightly.
In a medium bowl beat goat cheese; cream cheese, snipped fresh thyme, and ground black pepper until creamy. Spoon cheese mixture into a pastry bag fitted with a medium round tip. Pipe a small amount of cheese mixture into each date. Press date halves back together.
Cut sliced prosciutto into 3x1-inch strips. Wrap one strip tightly around each stuffed date. Place dates on prepared baking sheet.
Bake for 8 to 10 minutes or until prosciutto is crisp, turning after 5 minutes. Transfer to a platter. Serve immediately with toothpicks.