In a large mixing bowl and by hand, stir together the brown sugar, applesauce and oil until well combined.
Add both eggs and vanilla and whisk until well combined.
Set aside and combine dry ingredients well in a seperate bowl.
Stir dry ingredients into previously set aside wet ingredients just until combined. Do not overmix.
Stir carrot and pineapple into batter.
Spray your muffin pan very well with nonstick spray. Get the sides good so your muffins release well - yes, even if you use nonstick (I do too).
Using a 1/4 cup measuring tin for mini muffin pan or 1/2 cup tin for regular muffin pan, fill holes almost full and bake approx 15 minutes for mini muffins or 20 minutes for regular. Your oven may vary, but a toothpick inserted into middle muffin hole should be moist, with a few crumbs, but not sticky wet.