In electric mixer, beat butter and both sugars until combined and creamy. Add egg and vanilla. Mix well.
Add flour and salt and mix until soft dough forms.
Shape dough into 1-inch balls and place on greased baking sheet. Use your thumb to make an indent in the center of the dough ball. Chill cookies in fridge for 20-30 minutes.
Bake cookies for 10-12 minutes. If centers puff up, gently push down with spoon to form indent again. Cool completely.
In microwave-safe bowl, add caramel bits and water. Microwave for 20 seconds, stir, microwave, stir until melted and smooth.
Fill the center of each cookie 3/4 of the way with the caramel and sprinkle flaked sea salt on top. Let cookies sit for 30 minutes to allow caramel to set up.