Prepare the Cupcakes

Preheat oven to 325°F. Line a 12-cup muffin tray with 12 paper liners; set aside.

Whisk together flax meal and hot water in a medium bowl until combined. Add brown sugar, granulated sugar, and applesauce, and whisk until combined; set aside.

Combine flour, pumpkin pie spice, baking powder, and salt in a medium bowl. Add flour mixture to flax mixture, alternately with oil, stirring to combine after each addition. Fold in shredded carrots until evenly distributed. Spoon batter evenly into prepared liners; about 1/3 cup in each liner. Bake in preheated oven until a wooden pick inserted into centers of Cupcakes comes out clean, about 40 to 45 minutes. Transfer pan to a wire rack to let cool completely, about 1 hour.

Prepare the vegan buttercream

Stir together aquafaba, sugar, cream of tartar, and xanthan gum in a medium saucepan over medium-low heat until sugar is dissolved, stirring often, about 4 minutes. Transfer mixture to a stand mixer fitted with a whisk attachment and beat on medium-high speed until slightly thickened and cooled slightly (about 5 minutes). Reduce speed to low and gradually add softened butter, about 2 tablespoons at a time, until combined. With mixer still running on low speed, add salt, lemon juice, and vanilla. Increase mixer speed to medium-high and beat until mixture is thick, glossy, and holds its shape, about 10 to 15 minutes. ( Mixture may break or appear grainy after a couple of minutes; keep whisking and it will come together and become glossy.) Spoon or pipe buttercream onto cooled Cupcakes and serve.