Preheat oven to 350 degrees F. Line baking sheets with parchment paper.
Place the butter in a medium skillet over medium heat. Melt completely. Continue to cook, whisking frequently, until the butter foams, bubbles slightly, and begins to brown. Continue whisking until the butter is evenly brown, being careful not to burn. Remove from heat and allow to cool.
While the butter cools, in a medium bowl, whisk together the flour, baking soda, and salt. Set aside.
In the bowl of an electric mixer, combine the sugars and brown butter, mixing on medium speed until well blended and smooth.
Blend in the egg and yolk, scraping the bowl as needed. Blend in vanilla and yogurt or sour cream. With the mixer on low, blend in the dry ingredients until just incorporated.
Use a medium dough scoop to place a ball of dough onto the prepared pan (about 2 tablespoons). Press the center of the dough down to create an indentation. Place a caramel in the indentation and pinch together the edges of the dough over the caramel to completely hide it in the center of the dough. At this point, you may pinch the top of the dough in multiple spots to create a more rustic look. Repeat with remaining dough, spacing cookies at least 2 - 3 inches apart. Chill dough on cookie sheets in refrigerator for 30 minutes.
Bake the cookies until golden brown, about 14 minutes, rotating the pans halfway through cooking. Let cool briefly on the cookie sheet, then transfer to a wire rack to cool completely. Serve slightly warm to achieve gooey caramel texture.