Whisk the flour and salt in a bowl. Beat the butter, cream cheese and sugar in a large bowl with a mixer on medium speed until fluffy, 3 minutes. Beat in the egg, lime zest and extract. Reduce the mixer speed to low; beat in the flour mixture. Transfer half of the dough to a separate bowl; tint with food coloring.
Dust the dough with flour; roll out each color separately on floured parchment into a 1/4-inch-thick rectangle. Freeze until firm, 15 minutes.
Trim each piece into a 4 1/2-by-7 1/2-inch rectangle, reserving the scraps. Cut each rectangle lengthwise into nine 1/2-inch-wide strips. Lay 3 strips side-by-side on a baking sheet (white, green, white). Set 3 more strips on top, alternating colors (green, white, green). Top with three more strips (white, green, white). Build another log starting with green-white-green. Chill the logs.
Gather the scraps, keeping the colors separate. Roll out each on floured parchment into a 5-by-8-inch rectangle, 1/4 inch thick. Wrap the green rectangle around the log with the white corners and wrap the white rectangle around the log with the green corners; trim the excess. Wrap the logs in plastic; chill 2 hours.
Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment. Slice the logs into 1/3-inch-thick cookies; arrange 1 1/2 inches apart on the baking sheets. Bake, switching the pans halfway through, until set, 12 to 14 minutes. Let cool 5 minutes; transfer to racks to cool completely.