Preheat oven to 350 degrees Fahrenheit.
In the bowl of a stand mixer fitted with the paddle attachment, combine the butter and sugars. Cream until light and fluffy. Add the egg and vanilla. Mix well and scrape the bowl with a spatula.
In a separate bowl, stir together the flour, salt, and baking soda. Add dry ingredients to the butter mixture until just combined, about 1 minute.
Add the salted caramel baking chips (save some of each to top cookies with, if desired). Mix gently on low speed until just combined, about 30 seconds. *If dough is crumbly, add 1 Tbsp of milk and mix gently until dough comes together.
Line a baking sheet with a silicone baking mat, parchment paper, or spray with cooking spray. Scoop cookie dough balls and place onto prepared baking sheet about 2-inches apart. (12 small cookies per tray or 8 large cookies per tray.)
Unwrap caramels and cut each caramel in half. Press half of a caramel into the center of each cookie dough ball until it's level with the cookie dough. (Top with some of the saved salted caramel baking chips, if you set some aside.)
Bake at 350 degrees Fahrenheit for 8-10 minutes depending on size, until edges are just barely starting to brown. Immediately sprinkle the top of the hot cookies with coarse salt. Allow to cool for 1 minute on tray, then transfer to a wire rack to cool completely. Store in an airtight container at room temperature up to 1 week.