In a stand mixer, combine butter and sugar and beat until light and fluffy, about 2-3 minutes.
Add in egg and vanilla and beat until combined.
Add in flour, baking soda and salt. Mix on low until combined.
Shape dough into a disc and wrap in plastic and chill for one hour or more.
Roll dough into balls, about 1 inch and place on a parchment covered baking sheet. Use end of thumb or round end of wooden spoon and push a hole into the ball. Make a deep well but don't go all the way through to the baking sheet.
Bake for 12 minutes or until golden brown on bottom. Cool on cooling rack
Caramel Topping
Combine brown sugar and butter in a small saucepan and cook over medium heat. Bring to a boil and stir occassionaly
Boil for one minute and then remove from heat and add in whipping cream. Stir to combine and allow to cool.
Once cookies have cooled and caramel has cooled and set up partly, spoon caramel into the middle of the cookie. Allow caramel to set up and then sprinkle with coarse salt.