These donuts are a labor of love and patience as they require leaving time to rise, but your kitchen will smell like a bakery when it comes time to cook them.
In a large bowl, combine warm water and yeast.
Let it sit 5 minutes or until foamy.
Whisk in 1 cup of the flour until smooth, then cover the bowl tightly with plastic wrap.
Place a tea towel over the bowl and let it sit for 30 minutes. The batter should look bubbly one the time has passed.
Meanwhile, in a medium bowl, use and electric mixer to beat together the butter and sugar for 2 full minutes until pale and fluffy. Add the eggs one at a time and beat in. Beat in the vanilla and salt.
Whisk this butter mixture into the yeast/flour mixture until fully incorporated.
Then, begin beating in the remaining 3 1/2 cups of flour. Beat on medium speed, using a hook attachment until the dough wraps around the hook and no longer sticks to the bowl. If you are mixing by hand,using a wooden spoon, mix until the dough sticks to the spoon and not the bowl.
Cover the bowl in tightly wrapped plastic, cover with a tea towel, and let sit at room temperature to rise until about tripled in size, approximately 2 hours.
Punch the dough down and then wrap it tightly in plastic wrap. Place the wrapped dough in a large plastic ziplock bag and refrigerate for at least 3 hours, or up to 16.
Remove the dough from the plastic bag and wrap, divide in half, and roll out to 3/8 inch thickness on a well floured surface.
Cut, using a doughnut or biscuit cutter, into rounds.
Repeat with the other half portion of dough.
If the dough is a bit sticky; sprinkle a bit of flour over the top and work it in with your hands so that the dough is rollable. Let rise for 1 hour.
Once you are ready to fry the donuts, pour about 2 1/2 inches of oil into a large heavy duty pot. Be sure the pot is large enough that you have 2-3 inches of space at the top of the top without oil in it so it dosen't splash over.
Bring oil to somewhere between 350 and 375 degrees in a large, heavy duty pot.
Fry doughnuts in small batches, being sure not to crowd the pot. They should fry for about 1-2 minutes per side, and should be golden brown in color.
Take one doughnut out and cut into it to see if it's cooked.
Remove cook donuts with tongs and place onto a wire rack lined with paper towels. Repeat until all doughnuts are fried.
For the ganache:
Place chocolate into a heatproof bowl.
In a medium saucepan, heat cream over medium heat until steaming. Don't boil the cream.
Pour the hot cream over chocolate, let sit for one minute, then whisk until smooth.
Drizzle over the donuts while it is warm.
Garnish with cake sprinkles.