For the doughnut holes: Heat 3 to 4 inches of oil to 350 degrees F in a Dutch oven or medium pot fitted with a deep-fry thermometer over medium heat. Line a rimmed baking sheet with paper towels.
Stir together the flour, sugar, baking powder, salt, cinnamon and cardamom in a large bowl until combined. Add the milk, egg, egg white and melted butter and whisk until well combined. Set aside while you prepare the finishing.
For finishing: Place about 1/2 cup sugar each in individual small bowls--1 for each color. Use a toothpick to add a small amount of your desired food coloring to its own bowl, then stir together with a fork to combine as best you can. Reach in with your fingers and continue to mix until fully incorporated. Set aside.
For frying: Use a small ice cream scoop (no larger than 1 tablespoon) to scoop balls of batter into the hot oil. Working in batches, fry the doughnut holes until very golden brown and fully cooked through, 3 to 4 minutes. Remove the doughnut holes from the oil with a slotted spoon and transfer to the prepared baking sheet. Repeat with the remaining batter. While the doughnut holes are still warm, toss them in the colored sugar to coat well.
Transfer the jelly to a pastry bag fitted with a medium round tip. Stick the pastry tip into a doughnut hole and squeeze in some of the jelly until it feels heavy. Repeat with the remaining doughnut holes. Serve immediately.