In the bowl of a stand mixer fitted with the paddle attachment, beat butter, sugar, and salt at medium speed until light and creamy, 2 to 3 minutes, stopping to scrape sides of bowl. Beat in egg until combined; scrape sides of bowl. Beat in vanilla; scrape sides of bowl. With mixer on low speed, gradually add 2½ cups (313 grams) flour, beating until just combined.

Divide dough among 3 bowls (about 260 grams each). Add preserves and remaining 2 tablespoons (16 grams) flour to first bowl, gently kneading or stirring with a rubber spatula until combined; tint with pink food coloring as desired.

Add cocoa to second bowl, gently kneading or stirring with a rubber spatula until combined.

Add pistachios and almond extract to third bowl, gently kneading or stirring with a rubber spatula until combined; tint with green food coloring as desired. Shape each dough into a 4½x4-inch rectangle (¾ inch thick). (The size is important; try to keep the sides as straight and squared as possible.) Wrap each in plastic wrap, and refrigerate until firm, at least 2 hours, preferably overnight.

Trim sides of chilled doughs to straighten. Slice each slab into 6 (4x¾-inch) strips. Lay down one of each color (brown, green, and pink), gently pressing to adhere. Using a pastry brush, lightly brush egg white on surface, and gently press strips together to adhere. Repeat with 3 more alternating strips and more egg white. Top with final layer of alternating strips. Gently press sides of log to adhere everything. Repeat with remaining strips to give you 2 (4-inch-long) logs. Wrap each in plastic wrap, and refrigerate until cold, at least 2 hours.

Preheat oven to 350°F (180°C). Line baking sheets with parchment paper.

Using a sharp knife, trim ends of logs. Cut logs crosswise into ½-inch-thick slices, and place 2 inches apart on prepared pans.

Bake, one pan at a time, until edges are golden brown, 10 to 15 minutes. Let cool on pan for 5 minutes. Remove from pan, and let cool completely on a wire rack. Store in an airtight container for up to 1 week.