Prepare the pastry cream first and chill it completely in the refrigerator.
Warm 1 cup of whole milk until it is just warm to the touch, around 105-110F.
Stir 3 tablespoons of active dry yeast and 1 tablespoon of sugar into the warm milk. Let it sit for a few minutes until bubbles form.
In a large bowl, combine the yeast mixture with 1/2 cup granulated sugar, 1/4 cup shortening, 1 egg, 1 egg yolk, and 1 and 1/2 teaspoons vanilla extract. Stir well.
Add 3 and 1/4 cups of bread flour, 2 and 1/2 teaspoons kosher salt, 1/2 teaspoon baking powder, and 1/2 teaspoon baking soda to the wet ingredients.
Knead the dough using a dough hook or by hand for 5 minutes until it is stretchy and elastic. Add a little more flour if the dough is too sticky.
Grease a large bowl, place the dough ball inside, and turn to coat. Cover with greased plastic wrap.
Create a warm proofing environment in your oven by heating it briefly to 350F, then turning it off. Place a pan of boiling water on the bottom rack.
Place the covered bowl of dough on the top rack of the warm oven and let it rise for 1 hour, or until doubled in size.
Prepare three baking sheets with parchment paper squares.
Turn the risen dough onto a lightly floured surface and knead briefly. Roll it into a 1/2 inch thick circle.
Cut out donut shapes using a 2 and 1/2 inch biscuit cutter, pressing straight down without twisting. Place each donut on a parchment square.
Return the shaped donuts to the warm, turned-off oven with a pan of boiling water and let them rise for 30-45 minutes until doubled.
Heat about 2 inches of canola oil in a high-sided skillet to 350F.
Prepare a station with paper towels and cooling racks for the fried donuts.
Carefully slide each donut into the hot oil using the parchment paper. Fry for 30-60 seconds per side until golden brown.
Remove donuts with tongs and place them on the cooling rack. Check and maintain the oil temperature between batches.
While the donuts are still warm, roll them in granulated sugar.
Let the sugared donuts cool completely.
Prepare the filling by whipping 3/4 cup heavy cream with 2 tablespoons powdered sugar until stiff peaks form, then gently fold in the whisked pastry cream.
Fill the cooled donuts using a piping bag or a ziplock bag with the end snipped off, inserting the tip into a hole made in the side of each donut.
Fill donuts with the cream filling just before serving, as they do not store well overnight.
Store unfilled donuts in an airtight container or a loosely closed paper bag.
Store leftover pastry cream with plastic wrap pressed directly onto the surface in the refrigerator.
Cool leftover frying oil completely, strain it into a container, and store for future use.