Add both sticks of butter to a small saucepan and set over medium heat. Swirl the pan as the butter melts and starts to foam, a few minutes. With a spatula, constantly stir the butter until the solids brown and it takes on a fragrant popcorn-like aroma. Transfer to a large bowl to cool to room temperature.

If you’re using quick oats, whisk all of your dry ingredients together in a bowl. If using old-fashioned rolled oats, add all of the dry ingredients to a food processor and pulse it a few times to break the oats apart into a crumbly mixture. Set aside.

Add the sugars, eggs, and molasses to the bowl with the cooled brown butter and whisk together until thick and gooey. Pour in the sourdough discard and vanilla extract and whisk to combine.

Dump the bowl of dry ingredients into the bowl of wet ingredients. Use a spatula to fold and mix the thick cookie dough together just until no floury spots remain.

Preheat the oven to 350ºF (177ºC) and line baking sheets with parchment paper. Use a 2 TBS cookie scoop or spoon to portion out about 26-28 equal-sized cookies and space them a few inches apart on the baking sheets. Bake for 13-15 minutes, or until the cookies spread and the edges just start to brown. Cool on the baking sheets for 5 minutes before transferring to wire racks.

While baking and cooling, make the cream cheese filling. In a mixing bowl, combine the softened cream cheese, butter, salt, and vanilla extract until light and fluffy. Add in the powdered sugar and mix until thick and creamy. Adjust consistency with cream or more powdered sugar as needed.

Spoon or pipe a couple of tablespoons of cream cheese filling to the center of a cooled cookie and sandwich with another cookie. Repeat with the remaining cookies.