Start by beating 8 ounces of softened cream cheese in a mixing bowl until fluffy and smooth. Gradually add 1 cup of powdered sugar, 2 tablespoons of all-purpose flour, 1 teaspoon of vanilla extract, and a pinch of salt. Combine until fully blended. Next, scoop tablespoon-sized portions of the mixture, shape them into balls, and place them on a baking sheet lined with parchment paper. Freeze these filled balls for at least 30 minutes until firm.

In a large bowl, whisk together 2 cups of all-purpose flour, 2 tablespoons of cocoa powder, ½ teaspoon of baking soda, and a pinch of salt. In a separate pan, brown ½ cup of unsalted butter over medium heat until it turns golden and emits a nutty aroma; this should take about 5 minutes. Once browned, remove the butter from heat and mix in 1 cup of brown sugar, ½ cup of granulated sugar, 1 teaspoon of vanilla extract, 1 cup of sourdough starter, and 1 egg yolk. Add a few drops of red food coloring to achieve your desired hue.

Pour the cooled brown butter mixture into the dry ingredients. Stir everything together until just combined, being careful not to overmix. The Sourdough Red Velvet Cookie dough should be smooth yet slightly thick. If it feels too sticky, you can chill it in the refrigerator for about 15 minutes. Preheat your oven to 350°F (175°C) while you prepare to assemble the cookies.

Using a cookie scoop, take a generous scoop of dough and roll it into a ball. Flatten it slightly in the palm of your hand, creating a small indent in the center. Place a frozen cream cheese filling ball into the indent and wrap the dough around it, ensuring the filling is completely sealed within the dough. Repeat this process until all the dough and filling are used.

Place the assembled cookies on a parchment-lined baking sheet, spacing them about 2 inches apart. Bake the Sourdough Red Velvet Cookies in your preheated oven for 12 minutes, or until the edges are set and the tops are just starting to crack. Allow them to cool on the baking sheet for 5 minutes, then transfer the cookies to a wire rack to cool completely.

Once the cookies have cooled, melt a handful of white chocolate chips in a microwave-safe bowl, heating in 30-second intervals until smooth. Drizzle the melted white chocolate over the cooled Sourdough Red Velvet Cookies using a fork or piping bag for a decorative touch. Let the chocolate set briefly before serving to your family and friends.