Warm the Greek yogurt. Place the nonfat Greek yogurt in a microwave safe bowl and microwave for 30 seconds until slightly warm to the touch. Do not overheat. Cold yogurt will slow the starter significantly and extend your fermentation time.
Combine wet ingredients. In the bowl of a stand mixer fitted with the dough hook, combine the active sourdough starter, warmed Greek yogurt, eggs, granulated sugar, and vanilla extract. Mix on low speed until just combined.
Add dry ingredients. In a separate bowl, whisk together the bread flour, baking powder, fine sea salt, and ground cinnamon. Add the dry ingredients on top of the wet mixture. Cut the softened butter into small pieces and add to the bowl.
Mix the dough. Mix on low speed for 2 minutes until a shaggy dough forms and the flour is fully incorporated. The dough will feel sticky at this stage. This is normal for a yogurt based sourdough dough.
Rest the dough. Cover the bowl and let the dough rest for 30 minutes. This hydration rest is essential. Do not add flour before the rest period is complete.
Knead. After the rest, mix on low for 1 minute to bring the dough back together, then increase to medium low and knead for 6 to 8 minutes until the dough is smooth, soft, and slightly tacky. It should pull away from the sides of the bowl. It will be tacky. Only add bread flour in 10 gram increments after the full knead if the dough is still sticking to the sides.
Bulk ferment. Transfer the dough to a lightly oiled bowl and cover. Let rest at room temperature for 1 hour. After 1 hour, perform one set of stretch and folds with lightly dampened hands. Cover and continue to ferment for 5 to 6 more hours until the dough looks noticeably puffed and airy. Total bulk ferment is approximately 6 to 7 hours, however the environment of your kitchen can slow or speed up the process. Watch the dough, not the clock.
Divide and shape. Turn the dough out onto a lightly floured surface. For ring donuts, gently roll the dough to 3/4 inch thickness and cut rounds using a 2.5 to 3 inch cutter. Use a 1 inch cutter to cut the center holes. For solid rounds, divide the dough into 100 gram portions and roll each into a tight smooth ball. Place each shaped donut on its own small square of parchment paper.
Second proof. Cover loosely and proof at room temperature for 1 to 3 hours. In a kitchen at 74 degrees, 2 hours was my sweet spot. If your kitchen runs warmer, check at 1 hour. If it runs cooler, give them the full 3 hours. The donuts are ready when visibly puffed. To test, place one donut gently in the oil. If it floats immediately the whole batch is ready to fry.
Heat the oil. Pour avocado oil into an electric fryer, heavy bottomed pot or Dutch oven to a depth of 2 to 3 inches. Heat slowly to 350 degrees F using an instant read thermometer to monitor the temperature. While heating up mix your glaze(s), if using.
Make the chocolate glaze. Whisk the powdered sugar and cocoa powder together first to remove lumps. Add the melted butter, vanilla extract, and pinch of salt. Whisk in the whole milk one tablespoon at a time until the glaze is smooth and glossy.
Make the vanilla glaze. Whisk the powdered sugar, vanilla extract, vanilla bean pulp(if using), salt, and whole milk together until smooth and pourable. Add milk gradually until you reach a consistency that coats the back of a spoon.
Fry the donuts. Lower each donut into the oil parchment side down using tongs. Within about 10-15 seconds the parchment will release on its own and can be removed with tongs. Fry 2 to 3 donuts at a time. Do not crowd the pot. Fry for 1 minute 20 seconds to 1 minute 30 seconds per side until deep golden brown. Transfer to a wire rack or paper towel lined sheet to drain. Check the oil temperature between batches and let it return to 350 degrees F before adding the next round.
Glaze the donuts. Let the donuts rest for 3 to 5 minutes after frying before dipping. Dip the top of each donut into the glaze of your choice and let the excess drip off. Set on a wire rack. Add sprinkles or toppings immediately before the glaze sets. The glaze will begin to set and harden within about 10-15 minutes.