Whisk together flour, baking powder, and salt in a large bowl. Cream butter and sugar with a mixer on medium speed until pale and fluffy. With mixer running, add egg and vanilla. Reduce speed to low. Add flour mixture gradually, beating until just incorporated. Divide dough in half; flatten each half into a disk, and wrap each in plastic. Refrigerate until firm, at least 1 hour (or overnight).
Preheat oven to 325 degrees with racks in top third and lower third. Let 1 disk of dough stand at room temperature just until soft enough to roll, about 10 minutes. On a lightly floured work surface, roll out dough to just under 1/4-inch thickness, adding more flour as needed to keep dough from sticking. Cut out cookies with a 2-inch heart cutter, and place them 2 inches apart on a parchment-lined baking sheet. Roll out scraps once, and repeat. Repeat with remaining disk of dough. Freeze cookies until very firm, about 15 minutes.
Bake cookies, rotating sheets halfway through and switching from top to bottom, until edges turn golden brown, 14 to 16 minutes. Let cool on sheets on wire racks. Cookies will keep, covered, for up to 1 week.
Divide royal icing into 1/2-cup portions in small bowls. Tint each with a different gel-paste food coloring, starting with just a drop, mixing well, and adding more, drop by drop, to reach desired shade. Transfer 1 bowl of icing to a pastry bag fitted with a 1/8-inch round plain tip. Pipe the outline around edge of each cookie, then fill in with frosting. Poke air bubbles with a toothpick. Transfer decorated cookies to a parchment-lined baking sheet, and let stand uncovered overnight until dry.
Arrange stamp letters to create desired phrases. (We stamped LUV U, UR A QT, BFF, LYLAS, B MINE, and 2GTBT.) Fold a paper towel into quarters. Squeeze a small amount of red gel-paste food coloring onto a paper towel, and press stamp in coloring. (You may need to blot stamp a few times on a clean paper towel if coloring is too thick.) Lightly press stamp on top of icing in center of each cookie. Let stand until dry, about 30 minutes. Stamped cookies will keep, covered, for up to 3 days.