Preheat the oven to 350°F. Grease a 9x13 baking pan and set aside.
Cream together 1/2 cup unsalted butter and sugars (1/2 cup granulated sugar and 1/2 cup brown sugar) until light and fluffy. This can be done by hand or in a stand mixer with the paddle attachment. Mix in 1 1/2 cups applesauce2 large eggs, and 1 teaspoon vanilla extract.
In separate bowl, combine 2 cups all-purpose flour, 1 1/2 teaspoon baking soda, 1 1/2 teaspoon cinnamon, 1 teaspoon baking powder, and 1/2 teaspoon kosher salt. Mix into wet ingredients until well combined, scraping bowl as needed. Stir in 1/2 cup pecans.
Transfer to prepared baking dish and bake in preheated oven 35-40 minutes or until inserted toothpick comes out clean. When done, remove from oven and allow to cool on wire rack.
To make the brown butter frosting:
While the cake is cooking, make the frosting. In a saucepan over medium heat, melt the 1/2 cup unsalted butter. Whisk frequently, scraping the bottom of the pan. As the butter foams, continue whisking and scraping to release any brown bits that may stick. Once the butter solids have sufficiently browned and begin to smell nutty, remove from heat. Take care to not burn.
Using the whisk attachment on a stand mixer, beat browned butter with 2 cups confectioners sugar on medium speed. Add 1 tablespoon vanilla extract.
Continue to mix and add 2 tablespoons milk or half and half, one tablespoon at a time, until desired consistency is achieved and frosting is smooth.
Frost over cooled cake and sprinkle with remaining 1/2 cup pecans.