Preheat oven to 350 degrees and line cupcake pan with liners. Set aside.
In a large bowl, cream together butter and sugar. Add egg and vanilla and mix until combined.
In a separate bowl, whisk or sift together the dry ingredients: flour, salt, cinnamon, ginger, nutmeg, baking powder, and baking soda.
Add half of the dry ingredients to the butter mixture and stir until just incorporated. Add half of the applesauce and mix until just incorporated. Repeat once more until all ingredients are incorporated into batter. Do not overmix.
Pour batter into cupcake liners, filling each ⅔ full. Bake in preheated oven for 20-22 minutes, or until a toothpick comes out clean.
Remove pan from oven and allow to cool for 5 minutes before transferring cupcakes to a wire rack to finish cooling.
Browned Butter Frosting
In a small saucepan over medium heat, melt butter and stir constantly. Butter will begin to foam and boil. After 4 to 5 minutes you will see brown flecks appear in the bottom of the pan. Continue to stir until butter is golden and flecks are a medium brown color. Watch closely, butter will brown quickly. Entire process will take about 5 to 7 minutes.
Remove butter from heat and pour into a small bowl. Refrigerate for 10 to 15 minutes, or until butter has cooled to room temperature and beings to solidify.
Add browned butter to a large bowl and whip until creamy. Add confectioner's sugar, half at a time, mixing slowly to incorporate. Increase speed and whip for 2 minutes until fluffy.
Then, add vanilla extract and heavy cream. Whip another 2-3 minutes, or until frosting is fluffy.
Pipe or spread frosting with a knife onto cupcakes and serve.
Over-mixing your batter will cause your cupcakes to rise quickly, then sink in the middle. To prevent this, only mix each ingredient until it is incorporated into the batter. Try using a spoon instead of a hand mixer to incorporate the dry ingredients and applesauce.
Cupcakes will keep for 3-4 days at room temperature in a tightly sealed container.
Allow cupcakes to cool completely before removing from paper liners. If still warm, cupcakes will stick to the paper liners.
Allow cupcakes to cool completely before frosting. Buttercream frosting melts between 90-95 degrees, and will melt if frosted onto a warm cupcake.