Preheat oven to 350. Drain pineapple (reserve juice).
In large bowl, whisk dry ingredients. Make a well in center, add eggs, oil, vanilla and reserved pineapple juice.
Beat at medium speed until blended (about 1 1/2 min). With a spoon, fold in pineapple, applesauce, carrots. Spoon batter into baking cups. Bake approx 22 min, or until toothpick comes out clean.
Remove from oven, cool 2 minutes and transfer to rack.