In a large bowl using a hand mixer or the bowl of a stand mixer fitted with the paddle attachment, beat the softened butter, granulated sugar, and vanilla extract together on medium speed for 2-3 minutes, until light and fluffy. Add the eggs one at a time, beating well after each addition until fully incorporated.
In a separate medium bowl, whisk together the all-purpose flour, baking powder, and salt.
With the mixer on low speed, gradually add the dry ingredients to the wet ingredients, mixing just until a soft dough forms and no dry streaks of flour remain. Be careful not to overmix.
Remove about one-third of the dough from the bowl; this will be your colored "heart" dough. Place the remaining two-thirds (the plain dough) back in the bowl, cover, and set aside at room temperature. To the one-third portion, add several drops of natural food coloring. Mix thoroughly until the color is even, adding more as needed to achieve your desired pink shade.
Shape the pink dough into a disc, wrap it, and place it in the freezer for 15-20 minutes until firm but not rock hard. On a lightly floured surface, roll the chilled pink dough into a rectangle about ½-inch (1.25 cm) thick. Using a small heart cookie cutter (about 1.5 inches or 4cm), cut out as many hearts as possible. Lightly moisten one flat side of each heart with a dab of water and press them together to form a continuous log about 10 inches (25 cm) long. Place this heart log on a plate or tray and freeze for at least 30 minutes, until completely solid.
Take the reserved plain dough (which should still be soft and pliable). Divide it into two portions. Flatten each portion with your hands and firmly press it onto opposite sides of the frozen heart log, completely encasing the hearts and sealing any gaps. Gently roll the entire log on the counter to smooth it into a uniform cylinder, roughly 2 inches (5 cm) in diameter.
Wrap the finished log tightly in plastic wrap and refrigerate for at least 2 hours, or overnight. When ready to bake, preheat your oven to 350°F (175°C) and line baking sheets with parchment paper. Using a sharp knife, slice the cold dough into rounds just under ¼-inch (about ½ cm) thick. Place slices 2 inches apart on the prepared sheets.
Bake for 10-12 minutes, or until the cookies are set and the very edges are just beginning to turn a light golden brown. The centers will still look soft. Let cookies cool on the baking sheet for 5 minutes before transferring them to a wire rack to cool completely.