Preheat oven to 350°F (180°C). Line 3 rimmed baking sheets with parchment paper.
In the bowl of a stand mixer fitted with the paddle attachment, beat butter, 1½ cups (300 grams) sugar, and lemon zest at medium speed until fluffy, 2 to 3 minutes, stopping to scrape sides of bowl. Beat in egg. Beat in lemon juice and vanilla.
In a medium bowl, whisk together flour, salt, baking soda, and baking powder. Add flour mixture to butter mixture all at once; beat at medium-low speed just until combined, stopping to scrape sides of bowl.
Transfer half of dough (about 470 grams or 1½ cups) to a medium bowl. Add raspberries and desired amount of food coloring to remaining dough; beat at medium-low speed until combined, kneading together by hand toward end, if necessary.
In a small bowl, place remaining ¼ cup (50 grams) sugar.
Scoop lemon cookie dough by teaspoonfuls (about 7 grams each); scoop raspberry cookie dough by teaspoonfuls (about 7 grams each). Layer and press together 2 lemon dough scoops and raspberry dough scoops in an alternating fashion; roll into a ball, pinching together any seams, and roll in sugar. Place at least 2 inches apart on prepared pans. Repeat procedure with remaining dough.
Bake, one batch at a time, until edges and tops look just set and some small cracks are starting to form but centers seem puffed and slightly underdone, 8 to10 minutes, rotating pan halfway through baking. Let cool on pan for 5 minutes. Remove from pan, and let cool completely on wire racks.
Spoon Raspberry Jam Frosting into a large pastry bag fitted with a 3⁄8-inch round piping tip (Ateco #804). Pipe frosting onto flat side of half of cookies. Place remaining cookies, flat side down, on top of frosting.