Mix together butter and cornstarch until well combined. Add in powdered sugar and extract and mix until smooth. Add flour and mix until dough comes together and flour is all incorporated. Cover and refrigerate 10-15 minutes.
Preheat oven to 350 degrees F.
Scoop out about 1/2 a tablespoonful of dough. Roll into a ball and lay on parchment on a cookie sheet. Lay cookie dough balls out 4 across in 5 rows, fitting about 20 cookies per sheet. Bake 9 minutes. Remove from oven and flatten the tops of each cookie by gently pressing down with the bottom of a glass. Let sit for 2 minutes before transferring cookies to a cooling rack.
Melt butter. Stir in vanilla and milk. Whisk in powdered sugar. Stir until smooth and desired consistency is achieved. Let cool slightly before spooning onto cookies.
Let cool about 10 minutes. Spoon glaze on top. Transfer jam to a small bag and cut the corner to pipe 4-5 small drops of jam on top of glaze on each cookie. Use a toothpick to swirl the jam and the glaze.
Let cool to set. Store in an airtight container.
** If you live in a more humid climate or close to sea level, you will need more flour for this recipe. So instead of using just 1 cup, use 1.5 cups and bake 3-4 cookies initially as a test batch, to make sure the cookies don't spread too thin.