To make the browned butter, add the butter to a small saucepan over medium heat. Stir or swirl constantly until it begins to brown. Add the minced sage and continue to stir until butter is fully browned. Remove from heat and transfer to a small bowl. Set aside to cool.
Combine warm water, sugar and salt in a large bowl or bowl of a stand mixer. Stir until dissolved. Sprinkle with yeast. Let sit for 5 minutes until foamy.
Add flour, pumpkin purée and browned butter mixture to the bowl and stir until combined and a smooth dough forms, approximately 3-4 minutes. Remove the dough from the bowl, clean and grease the bowl, and replace the dough. Cover with plastic wrap and let sit in a warm place for 50-60 minutes, or until dough has doubled in size.
Preheat oven to 450°F. Line two baking sheets with parchment paper and set aside. Bring 10 cups of water and the baking soda to a boil.
As the water heats, transfer your dough to a clean surface. No need to flour or grease the surface -- you will need a bit of traction to form the dough. Divide the dough into 8 equal pieces; roll each piece out to a 12-inch rope. Form a U-shape with the rope, cross the ends over each other and push into the base of the U to create your pretzel shape. Place on the prepared pan.
Add each pretzel to the boiling water, makgin sure to dunk it under the surface, for 30 seconds each. Remove from the water with a large, slotted spatula and return to the prepared pan. Brush each boiled pretzel with egg wash - 1 egg yolk mixed with 1 Tbsp. water - and sprinkle with flaky sea salt.
Bake 12-14 minutes or until golden brown. Transfer to a wire rack to cool.