In a medium saucepan, melt the butter over medium-low heat. As the butter melts, it’ll start to foam, then turn a clear golden color. Continue to stir the butter, scraping up any browned specks from the bottom of the pan as it starts to brown. It should have a nutty aroma. Once browned and nutty, remove from the heat and pour into a large mixing bowl; cool slightly.
Add both sugars to the browned butter, whisking until completely combined. Whisk the pumpkin and vanilla into the mixture until smooth. Add the flour, salt, baking powder, baking soda, and spices, and use a wooden spoon to stir everything together until just combined. Stir in 3/4 cup of the white chocolate chips. Chill in the refrigerator for 1 hour.
Preheat oven to 350ºF. Press half of the dough into a greased 10-inch cast iron skillet. Spoon dollops of the dulce de leche evenly over the dough. Take chunks of the remaining dough, flatten slightly, and layer them side by side on top of the dulce de leche, sealing all of the edges together.
Bake for 27-30 minutes, or until edges are lightly golden. Immediately after removing from the oven, press remaining white chocolate chips into the top of the pizookie. Allow to cool for at least 45 minutes. Serve with a scoop of ice cream, and enjoy!