Preheat oven to 350°F. Butter a quarter sheet pan (9×13-inch) and line it with parchment paper.
In a medium bowl, whisk together flour, baking powder and ¼ cup yellow sprinkles. Set aside.
Combine butter, sugar, salt and lemon zest in the bowl of a stand mixer fitted with a paddle attachment. Cream on medium speed until light and fluffy, 3-5 minutes. Beat in eggs, one at a time, followed by vanilla extract and lemon juice. Scrape down sides of bowl. Add the dry ingredients and slowly mix until just combined.
Pour batter into prepared pan and use a spatula to spread it out into a smooth, even layer. Batter will be thick. Sprinkle surface with another ¼ cup yellow sprinkles. Bake for about 15 minutes, or until firm to the touch. Let cool completely, then freeze for 1 hour until cold but not rock hard.
Remove cookie bar from pan and cut it into two 9 by 6 ½-inch halves. Return to the freezer while you prep the ice cream.
Scoop ice cream into the bowl of a stand mixer fitted with a paddle attachment. Beat on low speed for about 30 seconds, until smooth, then add gumdrops and remaining ¼ cup sprinkles. Mix until combined.
Place one cookie slab sprinkle side down on the sheet pan. Working quickly, scrape ice cream on top and spread into a smooth, even layer. Place the remaining cookie half on the ice cream, with the sprinkle facing up, to make a giant sandwich. Gently press down and use an offset spatula to spread ice cream smooth on sides. Cover in plastic wrap and freeze until very firm, about 12 hours.
Carefully transfer ice cream cake to a large cutting board. Trim ¼-inch off of each edge and cut into 12 small sandwiches. Wrap each ice cream sandwich in plastic wrap or parchment paper and freeze again until firm.