4 tablespoons unsalted butter, plus more for slab or baking sheet
1 1/2 cups sugar
3/4 teaspoon peppermint extract or 6 drops oil of peppermint
Gel-paste food coloring, in pink
Liberally butter a marble slab or a rimmed baking sheet; place in the refrigerator until chilled. Meanwhile, in a heavy, small saucepan, combine 1/2 cup water with the butter and the sugar. Cook over medium-high heat until the butter has melted. Bring to a boil, reduce the heat to medium, and cover the pan for exactly 2 minutes to wash the sugar crystals from the side of the pan. Clip a candy thermometer to the side of the pan. Cook until the mixture reaches 260 degrees (the hard-ball stage).
Remove pan from heat. Quickly pour mixture all at once onto cold buttered marble slab or baking sheet, holding the pan close to the surface; don't allow the syrup to drip onto the surface of the candy. Let the candy stand undisturbed until bubbles no longer rise to the surface; this will take only about 1 minute. Using a buttered spatula, gently fold the edges toward center (this will soften edges).
When the candy is cool enough to handle, pour the flavoring over it and dab it with food coloring. Butter your fingers and pull the candy, keeping it in your fingertips and folding it over itself until the candy is opaque and satiny. Butter your fingers as necessary. Ridges will form in the pulled candy as it gets stiffer, and candy will look puffy and hold its shape. Threads of candy will form on stretched edges.
Pull candy into a rope about 3/4 inch in diameter. Using buttered scissors, cut candy into little pillows about 3/4 inch long. Place cut candy on a parchment-lined rimmed baking sheet; wrap tightly with plastic wrap to keep them airtight. Let rest 24 hours before serving. Store in an airtight container.