In a small pan, melt the vegan butter over medium heat. Once melted, remove the pan from heat and stir in the non-dairy milk, water, coconut milk, and sugar until dissolved.
In the bowl of a stand mixer fitted with the dough hook blade, combine the mixture with the sourdough starter and salt on low-medium speed to combine, about 1-2 minutes.
Reduce the speed to the lowest setting and add half of the flour (325g) plus the kosher salt and mix until the flour is incorporated. Repeat with another 325g of flour until the dough comes away from the sides of the bowl. Add more flour as needed, 2 tablespoons at a time. Mix for a total of 5-6 minutes until a soft smooth dough forms.
Turn the dough out onto a floured surface and knead it into a smooth ball, about 30 seconds. Put the dough into an oiled bowl and bulk rise until it is puffy and has doubled in volume, about 8-10 hours at room temperature. Make the dough in the evening to rise overnight and fry doughnuts in the morning. You can also refrigerate the dough once it has risen until you are ready to to proceed to shaping and proofing.
Turn dough out onto a lightly floured surface and roll it to about ½” thickness. Cut doughnuts using concentric circle cutters, about 2.5” circle for outer edge and 1“ circle for the center. Save centers to make doughnut holes. If making filled doughnuts do not cut out centers. Rework, roll and cut the dough scraps until all of the dough is used up.
Transfer doughnut cut-outs onto floured baking sheets. Cover with kitchen towels and proof in a warm spot for about 1 hour until puffy.
About 10-15 minutes before doughnuts are done proofing heat up the frying oil in a deep-sided 8” cast iron or enameled heavy-bottomed pan to 375˚F/190˚C.
Fry doughnuts in small batches until they are deep golden in color, about 45 seconds per side. If they are too delicate to pick up by hand, use a spatula to help pick them up and slide them into the oil. Use a slotted spoon, kitchen spider strainer or chopsticks to turn and lift doughnuts.
Transfer the fried doughnuts to cooling racks (or paper towels) set over sheet pans to allow the excess oil to drain. USE CAUTION when working with hot oil.
Sift powdered sugar into bowl and whisk in milk and vanilla paste or vanilla bean seeds to create a smooth glaze.
Dip doughnuts into glaze and return them to the cooling racks to let the glaze drip and set.
Warm coconut milk in a small saucepan, but do not boil. Remove from heat and whisk in chocolate chips, maple, and vanilla until the chocolate melts into a smooth glaze.
Dip doughnuts into glaze and return them to cooling racks. Add sprinkles or orange zest for decoration. The glaze will set as it cools.
Whisk together sugar and cinnamon in a bowl.
Roll the warm doughnuts in cinnamon sugar, coating all sides.
Pipe your choice of filling such as jam, custard or curd into the doughnuts and dust with powdered sugar.