Preheat oven to 350° F. Line 2 baking sheets with parchment paper; set aside.
Place 4 Tbsp of the softened butter in a large bowl and set aside. Meanwhile, in a small saucepan, heat the remaining 8 Tbsp (1 stick) unsalted butter over medium-low heat until melted. At this point, pay close attention because things move quickly and the butter will burn fast. The butter will melt and eventually sizzle, fizz and pop violently. Shortly past this point, it will begin to brown and you'll notice browned bits forming on the bottom of the pan, the butter will be turning amber in color and smell nutty and fragrant. Remove the pan from the heat, and pour the butter (and all the browned bits!) into the bowl with the softened butter. Combine the two carefully until the softened butter has melted as well.
To the butter bowl, add in both sugars and mix well. Add in the vanilla extract and pumpkin spice extract and mix well. Add in the egg and egg yolk, mixing well after each addition, followed by the flour, baking soda, baking powder, pumpkin pie spice and salt and mix until a soft dough comes together. Stir in the white chocolate chips.
Using a 1"-1.5" Tbsp size cookie dough scoop, portion out rounded balls of dough about 2" on the parchment-lined baking sheets. Bake for 10-12 minutes, rotating pans halfway through baking time to ensure even cooking. Cookies will appear slightly undone in the centers; this is okay as they will continue to set up as they cool on the baking sheets. Allow cookies to rest on the baking sheets for about 10 minutes before carefully transferring them to wire racks to cool completely.