Preheat the oven to 350 degrees F and line two cookie sheets with parchment paper. Set aside for now.
Using a stand mixer or an electric mixer, cream the butter, cream cheese, and sugar together until light and fluffy. Then add the egg, extracts, and lemon juice. Continue to mix on medium-high speed until light and fluffy and well-mixed.
Slowly add the flour and baking powder and mix until just combined. Use a spatula or wooden spoon to gently fold in the strawberries.
Refrigerate the batter for 5-10 minutes (or stick it in the freezer for 5 minutes), or until chilled. After, use a small cookie scoop to scoop batter onto the parchment-lined baking sheet.
Bake the cookies for 13-15 minutes, or until bottoms are just starting to brown and the centers are set. Rotate the cookie sheets halfway through baking to ensure even baking.
While the cookies are baking, remove the cream cheese and butter for the frosting to bring them to room temperature.
Remove cookies from the oven and place on a wire cooling rack to cool completely. Once completely cooled, make the frosting: use an electric mixer to combine the cream cheese and butter until light and fluffy. Add the remaining ingredients and mix well.
Take one of the cooled cookies, flip it upside down, and place a generous layer of frosting on top. Take another cookie and place it on top to form a cookie. Repeat with remaining cookies until all of them have been used.
Keep cookies refrigerated.
