Preheat the oven to 350°F (175°C). In a large mixing bowl, beat the softened butter with the dark brown sugar until the mixture is creamy, smooth, and thoroughly blended.

Add the egg and vanilla extract to the creamed butter mixture. Mix until both ingredients are completely incorporated.

In a separate bowl, sift together the all-purpose flour, baking soda, and salt. Gradually work this dry mixture into the butter mixture, stirring until an evenly combined cookie dough forms.

Prepare the filling in another bowl by mixing the chopped walnuts, 1/2 cup dark brown sugar, and sour cream. Stir thoroughly until the ingredients come together into a uniform, cohesive mixture.

Line a baking sheet with parchment paper or a Silpat baking mat. Using a 1 1/2-inch cookie scoop, portion the dough onto the prepared baking sheet.

Press your thumb gently into the center of each portion of dough to create an indentation. Spoon a generous teaspoonful of walnut filling into each hollow.

Bake in the preheated oven for 11–12 minutes, removing the cookies when the edges are just beginning to develop a light brown color.

Allow the freshly baked cookies to remain on the baking sheet for about 5 minutes. Transfer them carefully to a wire cooling rack and leave them there until completely cooled.

Once fully cooled, serve the cookies with milk or a preferred hot beverage. Keep any remaining cookies in an airtight container to help maintain their freshness.