Preheat the oven to 350 degrees F and grease two 9-inch round cake pans.
In a stand mixer, cream together the butter and sugar until light and fluffy. Mix in the vanilla. Add the eggs one at a time, beating after each addition. Mix in the pumpkin puree.
In a separate bowl, mix together the flour, baking powder, cinnamon, nutmeg, cloves and salt. Add the flour mixture to the pumpkin mixture and mix until thoroughly incorporated. Stir in the chocolate chips.
Divide the batter between the pans and bake until a toothpick inserted in the center of each comes out clean, 25 to 30 minutes. Let cool for 10 minutes before flipping onto wire racks to cool completely. Spread the Beurre Noisette Buttercream on the top of one of the cake rounds and top with the other.
Melt the butter in 1-quart saucepan over medium heat. Continue cooking, stirring occasionally and watching closely, until the butter turns golden brown, 5 to 7 minutes. Remove from the heat. Cool completely.
Combine the browned butter, confectioners' sugar and vanilla in a medium bowl. Beat at medi-um speed, gradually adding enough heavy cream for the desired spreading consistency.