In the bowl of a stand mixer fitted with a dough hook, combine flour, yeast, baking powder, and salt.
Add butter, milk, egg, egg yolk, honey, and vanilla.
Mix on low speed for 3 minutes.
Increase speed to medium for 8 minutes.
Place dough in a clean bowl that has been sprayed with cooking spray.
Let rise at warm room temperature (78° F) for 1 hour. If not using a proofing box with a water tray, cover bowl with plastic wrap.
While dough is proofing, line a sheet pan with parchment paper.
Spray lightly with cooking spray. Set aside.
Turn dough out onto a floured surface.
Sprinkle top with flour.
Roll to 1/2" thick.
Cover dough with a tea towel.
Let rest for 10 minutes.
Use a doughnut cutter to cut out doughnuts, gathering scraps and re-rolling as necessary.
Place on prepared pan.
Cover with a tea towel or plastic wrap that has been lightly sprayed with cooking spray.
Let rest for 15 minutes.
While dough is resting, heat 2" of oil in a large Dutch oven to 350° F.
Fry doughnuts in batches, 2-3 at a time, 1-2 minutes on each side, until golden brown.