Melt butter in a small saucepan over medium-low heat. Immediately pour into a heatproof bowl and set aside to cool slightly.
Over a small bowl place the pumpkin puree in several layers of cheesecloth or paper towels and press gently to remove the extra liquid until 1 cup of pumpkin puree gets reduced to ⅓ cup.
Preheat the oven to 350°F (180°C). Line two large baking sheets with parchment paper.
In a small mixing bowl, whisk together the flour, pumpkin pie spice, ground ginger, baking soda, and salt. Set aside
In the bowl of a stand mixer fitted with the paddle attachment, or in a large mixing bowl using an electric mixer, cream together the browned butter, brown sugar, and granulated sugar for 2-3 minutes until well combined.
Add the egg and vanilla extract and mix until fully combined. Then stir in the pumpkin puree until well combined.
Slowly mix in the dry ingredients into the wet, making sure to stop and scrape down the sides of the bowl as needed. Then gently stir in the butterscotch and chocolate chips until just combined.
Using a large cookie scoop, scoop the cookie dough and drop onto the prepared baking sheets. Very gently press down with your hand to flatten each ball of cookie dough slightly. Make sure to leave a little room between each ball of cookie dough as they will spread a little while they bake.
Bake the cookies at 350°F for 12-13 minutes or until the edges of the cookies are lightly browned and the top is set.
Remove from the oven and cool on the baking sheet for 5 minutes, then transfer the cookies to a wire rack to finish cooling.