Line 2 large baking sheets with parchment paper and lightly dust with powdered sugar. Set aside. In a large bowl, beat ½ cup softened butter until smooth and creamy using a handheld or stand mixer. Add 1 tablespoon heavy cream, ½ teaspoon peppermint extract, and ¼ teaspoon vanilla extract. Mix well. Gradually add 2 cups of powdered sugar, beating until smooth. Add the remaining 2 cups powdered sugar, 1 cup at a time, mixing until well combined. Scrape the bowl as needed. The dough should be stiff and not sticky (add more powdered sugar if needed for the right consistency). Divide the dough into separate bowls (one for each color). Add a scant drop of food gel coloring to each bowl and mix until the color is evenly distributed. Keep the remaining dough covered with a damp towel to prevent it from drying out. Pinch off small amounts of the dough (about 1½ tablespoons) and roll them into balls. Then, roll each ball into a ½-inch thick log and cut it into mini pieces, about ¼ to ½ inch in size. Place the pieces on the prepared baking sheets. Let the mints stand uncovered at room temperature for at least 3 hours (preferably overnight) until firm. Flip any mints with moisture on the bottom and allow them to firm up as well. Store the mints in a cool, dry place at room temperature for up to two weeks, or in an airtight container in the fridge for up to a month. They can also be frozen for up to 6 months. If freezing, store the mints in a single layer between sheets of wax paper to prevent sticking.