For the Beer Syrup:
Combine beer and sugar in a medium stockpot at medium-high heat.
Stirring regularly to avoid burning the sugar, bring the mixture to a boil.
Reduce heat to medium-low and simmer until thickened.
Cool completely before using.
For the Molasses Stout Cookies:
In a large bowl, whisk together flours, baking soda, and spices.
Cream together butter, brown sugar, and egg in a stand mixer until well blended. With the mixer on low, drizzle in the molasses and beer syrup unitl just combined.
Adding a few spoonfuls at a time, mix the dry ingredients into the wet ingredients on low speed.
Using a scoop, roll the dough into large balls. Dough will be very sticky.
Refrigerate dough balls for at least one hour before baking.
Preheat oven to 350 degrees. Line a sheet with parchment paper and place dough balls 2 inches apart onto the prepared sheet.
Bake for 8 to 10 minutes, until the tops are cracked and the edges are just starting to brown. Do not overbake.
Allow the cookies to cool on the pan for at least 5 minutes before removing to a wire rack.
For the Stout Infused Cream Cheese Frosting:
In a stand mixer, cream together the butter and cream cheese until smooth.
Slowly add powdered sugar & beat on high until smooth.
Add in vanilla and beer syrup, being careful not to over beat the frosting.