In a medium bowl, whisk 3 1/3 cups all-purpose flour, 2 teaspoons Nesquik Strawberry Milk Powder, 1 1/2 teaspoons baking powder, and 3/4 teaspoon kosher salt. In the bowl of a stand mixer fitted with a paddle attachment, combine 1 cup cubed unsalted butter and 1 cup granulated sugar. Beat on medium speed until light, fluffy, and doubled in volume, about 3 to 5 minutes, using a rubber spatula to scrape down the bottom and sides of the bowl occasionally. Reduce the mixer speed to low and add 1 large egg and 1 teaspoon strawberry extract, mixing until just combined. Scrape down the bottom and sides of the bowl with a rubber spatula. With the mixer on low, gradually add the dry ingredients. Increase the mixer to medium-low and beat until the dough clumps around the paddle and/or sides of the bowl, about 2 to 3 minutes. Scrape down the sides of the bowl with a rubber spatula. Tip dough into the center of a sheet of parchment paper around the size of a sheet pan and use a bench scraper to divide off 1/3 of the dough. Place the dough back in the stand mixer and add 2 to 3 drops of red food coloring; beat on low until pale pink.
Return to the reserved dough on the parchment paper. Place a second sheet of parchment paper over the dough, creating a parchment paper sandwich with the dough in the middle. Use a rolling pin to flatten the dough between the parchment paper sheets, working from left to right. Turn the dough 90 degrees and repeat — turning the dough every so often will help prevent the dough from cracking as you roll it. If the parchment paper starts to wrinkle and leave creases in the cookie dough, pull the sheet loose and smooth it before rolling the dough more. Continue until you have a sheet of dough between ⅛ and ¼-inch thick. Freeze for at least 5 minutes.
As the dough freezes, repeat the step above but with the dyed dough. Freeze for at least 5 minutes.
While the dough freezes, center a rack in the oven and preheat to 350 (F). Line two sheet pans with parchment paper.
Remove the undyed dough from the freezer and use cookie cutters to stamp out larger shapes from the dough. Use a smaller cookie cutter to stamp smaller shapes in the larger shapes, saving the cutouts for rerolling later.
Remove the dyed dough from the freezer and the smaller cookie cutter to stamp out smaller shapes. Nestle the smaller shapes into the larger cookies.
Place the the cookies on the prepared sheet pan, leaving at least 1½ inches between each cookie. Bake each sheet pan for 10 minutes, or until the cookies are golden brown around the edges. Cool the cookies on their sheet pan on a wire rack for 20 minutes, or until the edges and bottoms of the cookies have set and feel firm to the touch. The cookies can be kept at room temperature, in an airtight container or Ziploc bag, for up to 3 days.