Position oven racks in the upper and lower thirds of the oven and preheat to 350 degrees F.
Beat the butter and sugar in the bowl of a stand mixer fitted with a paddle attachment on medium-high speed until light and fluffy, about 5 minutes. Add the almond and vanilla extracts and egg and mix until combined. Reduce the mixer speed to low, add the flour and salt and mix until just combined.
Remove all but 1/4 cup of the dough to a flour-dusted surface and reserve. Add enough green gel food coloring to the dough left in the bowl to color it a vibrant green like a wreath, 3 to 4 drops. Mix on low until the dough is completely green. Remove the dough to a flour-dusted surface and gently fold until it comes together. Wipe the paddle and bowl of the stand mixer clean.
Place 1/4 cup of the reserved white dough back into the bowl. Add enough red gel food coloring to color it a vibrant red like a fire truck, 3 to 4 drops. Mix on low until the dough is completely red. Remove the dough to a flour-dusted surface and gently fold until it comes together.
Tear off 2 Ping-Pong ball-size pieces of the white dough and add to the barrel of a cookie press; add 1 marble-size piece of the green dough and 1 marble-size piece of the red dough. Repeat this pattern until the barrel is filled (you will not use all of the dough). Place a wreath disc onto the cookie press barrel according to the manufacturer's instructions. Press the cookies onto an ungreased baking sheet. Repeat 2 more times in the same pattern with the remaining dough.
Bake 2 of the batches until the cookies are firm but not yet browning on the bottom, 8 to 10 minutes. Cool the cookies for 1 minute on the baking sheets, then remove to a wire rack to cool completely, about 30 minutes. Bake the third batch and let cool. Store in an airtight container for up to a week.
When measuring flour, we spoon it into a dry measuring cup and level off excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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