Preheat oven to 375 degrees. Line a muffin tin with paper liners. In a mixing bowl, using an electric mixer, whip together cream cheese, 1/3 cup granulated sugar, lemon juice and 1/2 tsp vanilla on low speed until mixture is smooth, set aside.
In a mixing bowl, whisk together flour, baking powder, baking soda, cinnamon, salt and nutmeg, set aside. In a large mixing bowl, using an electric mixer on low speed, blend together 1 cup granulated sugar, buttermilk, vegetable oil, applesauce, eggs and 1 tsp vanilla for 30 seconds. Add in dry ingredients and mix just until combine. Using a rubber spatula, fold in grated carrots.
Fill paper lined muffin cups with an 1/8 cup muffin mixture (Note: it's important to keep these measurements even. Also, if you don't have an 1/8 cup measure out 2 Tbsp). Gently drop 1 Tbsp cream cheese mixture on top, in the center of each filled muffin cup, then cover the cream cheese mixture with an additional 1/8 cup muffin mixture. Sprinkle each muffin evenly with 1/2 tsp Raw Sugar. Bake 20 - 23 minutes until edges are lightly golden.
If necessary, run a knife around the edges of the cupcakes to remove from pan (note: I haven't tried this yet but spraying the top of the pan with non-stick spray before baking may help muffins remove more easily). Enjoy warm for a slightly runnier cream cheese filling or allow to cool 30-40 minutes for a set cream cheese filling. Store muffins in an airtight container.