Whisk together flour, baking powder, and salt. Beat together butter and sugar with a mixer on medium speed until pale and fluffy, about 3 minutes. Beat in egg and vanilla. Reduce speed to low, and gradually add flour mixture, beating until just incorporated.

Add beet juice, mixing and kneading to combine, until an even pink shade is reached. Flatten dough into a disk and wrap in plastic wrap. Refrigerate until firm, at least 1 hour and up to overnight.

Bring dough to room temperature, about 10 minutes, before rolling.

Preheat oven to 325 degrees with racks in top and lower thirds. Roll out dough on a lightly floured surface to just under 1/4 inch thick, adding more flour as needed to keep dough from sticking. Cut out even numbers of heart shaped cookies and place 1 inch apart on parchment-lined baking sheets. Freeze until very firm, about 15 minutes.

Bake for 8-10 minutes, rotating halfway through. Let cookies cool completely on baking sheets set on wire racks.

Meanwhile, make the filling. Beat butter with a mixer on medium-high speed until light and fluffy, about 3 minutes. Reduce speed to low, and gradually add sugar, beating until thoroughly combined. Beat in vanilla. If filling is too thick to spread, beat in milk, 1 tablespoon at a time, until spreadable.

Spread filling onto bottom side of half the cooled cookies, and sandwich with remaining cookies, pressing gently.