Heat 1/2 cup whole milk in a small stock pot. Add 1/2 cup bread flour when scaling and whisk for 30 seconds. Transfer to bowl of a stand mixer.
Heat stick of butter in a small sauce pan until browned. Remove from heat and add peeled cooked potato.
Transfer potato mixture to a bowl and mash until smooth. Add to the stand mixer and continue to mix on low.
Add the other 3 1/2 cups of bread flour, olive oil, 1 1/2 tbsp white sugar, 1/4 cup milk, maple syrup, 3 eggs, active dry yeast, and kosher salt.
Mix dough until fully cohesive and elastic (about 15 minutes). Remove and roll your dough into a large, tight ball.
Place in a greased bowl, covered, for 24-48 hours in the fridge.
Remove dough and split into twelve 110g portions. Roll each into a tight ball and transfer to a parchment-lined baking sheet.
Let rise, covered, for up to 1 hour, or until light and puffy, in a warm spot.
Heat 6 cups vegetable oil in a large dutch oven or deep sauté pan until it reaches 335°F. Prepare a sheet pan with paper towels and a wire cookie rack on top.
Fry donuts 3 at a time 3-5 minutes on a side until deep golden brown. Flip carefully and cook for 3-5 minutes on the second side.
Mix 1/3 cup white sugar with cinnamon in a large heat-proof bowl. Toss cooked donuts in the sugar mixture until coated on all sides.
Sprinkle the remaining sugar over the top of the coated donuts and brûlée with a kitchen torch, taking care not to burn.
Fill the donuts with a simple vanilla pastry cream, using a sharp metal tip to pierce the dough.