Brown Butter Hazelnut Layer Cake

Name: Brown Butter Hazelnut Layer Cake
Prep time: 2 h
Cook time: 30 min
Total time: 2 h 30 min
Cuisine: American, Southern
Ingredients: 6 ounces dry roasted hazelnuts (roughly chopped ); 1 cup granulated sugar; 1/4 cup water; 1-3 tablespoons vegetable oil (optional as needed); 2 cups cake flour (substitute all purpose flour); 1/2 cup hazelnut flour (substitute almond flour or all purpose flour); 2 teaspoons baking powder; 1/2 teaspoon salt; 1 cup brown butter (melted); 1 cup brown sugar; 1/2 cup granulated sugar; 3 large eggs (room temperature); 2 teaspoons vanilla extract; 1 teaspoon almond extract; 1 cup whole milk; 1 1/2 cups brown butter (room temperature ); 4 cups powdered sugar; 1 teaspoon vanilla extract; pinch salt; 2-3 tablespoons heavy cream (as needed)