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Whisk together the flour, baking powder, baking soda and salt in a medium bowl.
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Beat the butter and sugar with an electric mixer until fluffy, about 3 minutes. Add the egg and beat until combined, scraping down the sides as needed. Add the vanilla and beat to combine.
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With the mixer on low speed, gradually add the flour mixture and beat until incorporated. Transfer half of the dough to a medium bowl. Add 4 to 5 drops of the blue gel food coloring to the remaining dough and mix on low until combined.
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Use an ice cream scoop to drops spoonfuls of dough into a new bowl. Alternate the colors, until all the dough has been scooped. Cover and freeze until firm, about 20 min.
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Line baking sheet with parchment paper. Scoop balls of dough (¼ cup each) with a large ice cream scoop onto baking dish, spacing out 2 inches apart.
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Flatten each dough portion and a marshmallow half in the center. Fold up the cookie dough, pinching to seal, so the marshmallow sealed. Roll into a smooth ball and roll the top in the sprinkles. Repeat with remaining dough. Refrigerate until firm, about 15 minutes.
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Preheat oven to 375 F.
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Bake the cookies until lightly golden around the edges, 15 to 20 minutes. (If marshmallows cracks through the top, it will settle back down as it cools.) Let cookies cool 5 to 10 minutes on baking sheet.