Beat the butter and sugar on medium speed for about 1 minute, just until smooth. Beat in the egg, vanilla, and peppermint extract.

Add the flour and salt and mix on low until a soft dough forms. If crumbly, add cold water 1 teaspoon at a time.

Weigh the dough and divide it into two equal halves. Mix the red gel coloring into one half until evenly colored.

Shape each color into a block 6 inches long, 2 inches wide, and 2 inches tall. Wrap and chill for 1 hour.

Trim the rounded edges. Cut each block lengthwise into four ½-inch slabs, then cut each slab into four ½-inch strips, for 16 strips of each color.

Lay four strips side by side, alternating colors. Brush lightly with water and stack the next row with the colors flipped. Repeat to make a 4 by 4 checkerboard log. Build a second log with the remaining strips.

Press the sides firmly to close gaps. Wrap tightly and chill at least 2 hours or up to 4 days.

Heat the oven to 350F and line two baking sheets with parchment.

Trim the ends of each log, then slice into ¼-inch squares, turning the log a quarter turn every few slices. Place 1 ½ inches apart.

Bake 10 to 12 minutes, until the bottoms are lightly golden and the tops are pale. Cool 5 minutes on the sheet, then cool completely on a rack.

Beat the butter until creamy. Add the powdered sugar, peppermint extract, salt, and 1 tablespoon cream and beat 2 to 3 minutes until fluffy, adding more cream if needed.

Fold in the crushed candy canes and the optional drop of red gel coloring.

Pair the cookies by size. Pipe about 2 teaspoons of filling onto the flat side of half the cookies, top with the matching cookie, and press gently until the filling reaches the edges.