Begin by preparing the dough:
Warm milk to just above room temperature using the microwave or a saucepan. Pour into the work bowl of a stand mixer, gently stir in the boiled cider and the yeast. Let stand for 5 minutes.
After 5 minutes, use the batter attachment on your stand mixer to blend in the sugar, brown sugar, salt, vanilla extract, cinnamon, nutmeg, egg, egg yolks, and pumpkin puree. Mix on medium for at least 1 minute to be certain everything is evenly combined and smooth.
Turn off the mixer, switch to the dough hook, and add all of the flour at once. Start on low (to avoid the dreaded flour POOF) and gradually move up to medium high where it should stay for 4-6 minutes, or until you have a soft, sticky batter-like dough.
Butter a large mixing bowl or pan generously and scrape the dough into it. Cover with plastic wrap and let rise at room temperature until doubled in size (about 1 hour at average room temperature.)
After an hour, line 2-3 baking sheets with parchment or silpats and set them aside.
Generously flour your work surface and your rolling pin. Turn the dough out onto the counter and dust the top with flour. Knead for no more than 1 minute to deflate the dough.
Roll it out to about 1/2-an-inch thickness and use a 3-inch diameter round cutter to cut large doughnuts and a 1/2-1-inch thickness round cutter to cut the hole from the center. (Or use the small cutter to do a gigantic batch of doughnut holes!) Carefully transfer the cut doughnuts to the lined sheets with about 2-inches of space between each doughnut (or 1 inch between doughnut holes.)
Re-roll the scraps and cut more doughnuts. Repeat until you've used all the dough.
Cover the baking sheets with plastic wrap and let rise for about an hour. Take care not to let them rise too long or the yeast will exhaust its activity before it gets to the oven and you'll lose the light texture you're trying to get.
Preheat the oven to 400°F.
To Make the Brown Butter
While the oven preheats, melt the butter in a heavy-bottomed saucepan over high heat, swirling it to keep it from scorching. The butter will bubble and pop. As soon as the butter smells nutty and you can see lightly browned milk solids in the pan, remove the pan from a burner and set in a place to cool just slightly.
To Bake Doughnuts:
Bake the doughnuts in the preheated oven for about 20 minutes or until golden brown.
Bake the doughnut holes in the preheated oven for about 10 minutes, or until just golden.
To Assemble the Doughnuts:
Toss the granulated sugar and ground cinnamon together in a bowl. Set the cinnamon sugar next to the brown butter. Working with one doughnut at a time, dunk into the brown butter and flip to coat. Lift with a fork to allow the excess butter to drain away then drop into the cinnamon sugar. Toss to cover all surfaces with the cinnamon sugar, transfer to a plate, and repeat with the remaining doughnuts and doughnut holes.