In a large bowl, whisk together the warm milk, 1 tbsp sugar, and yeast. Let it sit for 5-10 minutes until foamy — this confirms your yeast is active.
In a large bowl, stir in whisk the pumpkin puree, eggs, milk, melted butter, vanilla, salt, pumpkin pie spice.
Add the flour, salt, remaining 3 tbsp brown sugar and stir until just combined, the dough will be sticky, shaggy. It should be much wetter than a traditional doughnut dough — resist the urge to add more flour to make it easier to handle. This stickiness is exactly what makes the no-knead method work.
Cover the bowl with plastic wrap and set it in a warm spot for about 1 to 1½ hours, or until doubled in size.
Generously flour your work surface and turn the dough out onto it. Dust the top with a little more flour so it's easier to handle, then gently pat or roll the dough to about 3/4 inch thickness. Cut out doughnuts using your cutter, gently re-rolling scraps as needed.
Transfer the doughnuts and holes to a parchment-lined baking sheet, spaced apart, and let them rise again for 15-30 minutes or until puffy.
Heat oil (to 350°F) in a heavy pot. Fry doughnuts in batches for about 60-90 seconds per side, until deep golden brown. Drain on a wire rack.
While still warm, brush doughnuts with melted butter and toss in the cinnamon-sugar mixture, or dip into the maple glaze and let set on a wire rack.