Place butter pieces in a medium skillet. Melt the butter over medium heat, swirling it in the pan occasionally. The butter will foam and pop, so be careful. Continue to swirl the pan often. Remove pan from heat once the butter starts to brown and smells nutty. There will be small brown bits on the bottom. The butter should be an amber color. Pour butter into a small bowl and cool to room temperature. You can put it in the fridge to speed up the process, that is what I do.
While the brown butter is cooling, whisk together the flour, baking soda, pumpkin pie spice, salt, and cinnamon in a medium bowl. Set aside.
In the bowl of a stand mixer, combine the browned butter with brown sugar and granulated sugar. Beat until creamy and smooth, about 3-4 minutes.
Add the eggs and vanilla extract and mix until combined. Beat in the pumpkin and mix until combined.
Turn the mixer to low speed and slowly mix in the dry ingredients. Mix until just combined. Stir in the oats and butterscotch chips. Chill dough in the fridge for 30 minutes.
Preheat oven to 350 degrees F. Line two baking sheets with Silpat baking mats or parchment paper and set aside.
When dough is chilled, form cookie dough into 2 tablespoon sized cookie dough balls. Place on prepared baking sheets, about 2 inches apart. Bake cookies for 12 minutes or until slightly brown around the edges. Remove from oven and allow the cookies to cool on the baking sheet for 2-3 minutes. Transfer to a wire cooling rack and cool completely.