1. Prepare the Cookie Dough Base: In a large mixing bowl, cream together the softened 1 cup butter and granulated sugar until the mixture is light and fluffy. This usually takes about 2-3 minutes with an electric mixer, and you’ll see the color lighten. Beat in the large egg and 1 teaspoon vanilla extract until well combined, then stir in the unfed sourdough discard. The mixture might look a little curdled at this stage, but don’t worry, it will come together.

2. Combine Dry Ingredients: In a separate medium bowl, whisk together the all-purpose flour, baking soda, and salt. Make sure these are thoroughly combined to ensure even leavening in your cookies.

3. Mix Wet and Dry Ingredients: Gradually add the dry ingredient mixture to the wet ingredients, mixing on low speed until just combined. Be careful not to overmix, as this can lead to tough cookies. Stop mixing as soon as no streaks of flour remain. The dough will be soft but manageable.

4. Shape and Bake the Cookies: Preheat your oven to 375 F (190 C) and line baking sheets with parchment paper. Drop rounded teaspoons of dough onto the prepared baking sheets, leaving about 2 inches between each cookie. These are mini cookies, so keep the dough balls small, roughly the size of a marble. Bake for 10-12 minutes, or until the edges are lightly golden and the centers are just set. They might still look a little soft in the middle, but they will firm up as they cool.

5. Cool the Cookies Completely: Remove the cookies from the oven and let them cool on the baking sheets for 5 minutes before transferring them to a wire rack to cool completely. It’s crucial that the cookies are fully cooled before filling, otherwise, the frosting will melt.

6. Prepare the Creamy Filling: While the cookies are cooling, prepare the filling. In a medium bowl, beat the 1/4 cup softened butter until creamy. Gradually add the powdered sugar, beating until smooth. Add 1 tablespoon of milk and 1/2 teaspoon vanilla extract, beating until light and fluffy. If the frosting is too thick, add the remaining tablespoon of milk, a teaspoon at a time, until it reaches a spreadable consistency.

7. Assemble the Cookie Sandwiches: Once the cookies are completely cool, spread a small amount of the prepared filling onto the flat side of one cookie. Gently top with another cookie, flat side down, to create a sandwich. Repeat with the remaining cookies and filling.