In a large bowl, beat together the sugar, powdered sugar, and butter using a hand mixer until light and fluffy (about 3–4 minutes). Add the oil and egg, and beat together until well combined.
In a separate bowl, whisk together the flour, salt, baking soda, and cream of tartar. Gradually add the dry mixture to the wet mixture, one cup at a time, mixing on low to medium speed until a soft dough forms. You may need to finish mixing by hand if the dough gets too thick.
Split the dough into three equal parts, placing each into a separate bowl.
In one bowl, mix ½ teaspoon cherry Jello with ½ teaspoon water, and ½ teaspoon berry Jello with ½ teaspoon water in a separate bowl.
Pour the cherry Jello mixture over one dough bowl and add red food coloring. Repeat with the berry Jello and blue food coloring for the second dough bowl. Leave the third dough bowl plain.
(Optional: Wear gloves to avoid staining.) Knead each colored dough until fully blended, adding more dye if needed.
Line a tray with parchment paper. Shape the plain dough into an 8–10 inch log and place it in the center of the tray. Shape the red dough into a log and place it on the left side of the plain dough. Repeat with the blue dough and place it on the right side.
Wrap the entire tri-color dough log in plastic wrap and refrigerate for 30 minutes.
Preheat the oven to 350°F and line a baking sheet with parchment paper. Remove the dough from the fridge and slice it crosswise into ¼-inch thick pieces.
Place a popsicle stick on the baking sheet and gently press a cookie slice on top. Use a fork to lightly press lines down the cookie to mimic the look of a Bomb Pop.
Bake the cookies for 10–14 minutes. Let them cool completely before serving. Enjoy!