In a small bowl, whisk the flour, salt and baking soda. In a large bowl, using an electric mixer at medium speed, beat the sugar and butter, occasionally scraping down the bowl, until light and fluffy, about 3 minutes. Beat in the egg and vanilla. Reduce to low; add the flour mixture and beat until just blended.
Divide the dough in half and place in 2 bowls. Stir the cocoa and milk into one bowl. Stir the peppermint into the other bowl. Divide the peppermint dough in half and place in separate bowls. Mix the red food color into 1 peppermint dough and the green into the other.
Divide the chocolate dough in half. Shape all the dough pieces into 3-by-5-inch rectangles, set on wax paper, and top with another sheet of wax paper. Roll out dough into 9-by-12-inch rectangles. Refrigerate the rectangles on baking sheets until firm, 1 hour.
Place the 2 chocolate rectangles on a work surface; remove top sheets of wax paper. Lightly brush with water. Remove the top sheet of wax paper from the green dough and invert onto 1 chocolate rectangle; repeat with the red dough onto the other chocolate rectangle. Roll just to seal, then remove wax paper.
Starting with 1 long side, curl the dough to begin forming a spiral. Using the bottom sheet of wax paper as an aid, roll the dough into a tight log, pressing to compact. Wrap each log in plastic; refrigerate until firm, 3 hours.
Position the racks in the upper and lower thirds of the oven; preheat to 350 degrees . Working with 1 log at a time, cut into 1/4-inch-thick slices. Place 1 inch apart on 2 parchment-lined baking sheets. Bake until the edges begin to brown, rotating the sheets halfway through baking, 10 to 15 minutes. Let cool on the sheets 1 minute before transferring to racks.