2tablespoonsmilk or heavy creamoptional, for thin icing (drizzle or dunk)
Making the dough
Heat the milk in the microwave or on the stove (in a pot over low/medium heat), stirring every 30 seconds until it starts to boil. Remove from heat and stir in the butter until it melts. Then, add the sugar and whisk to dissolve.
Check the milk mixture's temperature with a thermometer. If it's between 95-110°F, stir in the yeast. (If it's above 110°F, stir until it cools to that range. If it's below, microwave it for a few seconds) When the yeast starts to bubble (this should happen in about 5-10 minutes), mix in the egg yolk, salt, and vanilla.
Using a mixer with a hook attachment (or kneading with your hands), mix in ¾ cup of flour. The dough should just barely be sticky to the touch. If it's still really sticky, gradually mix in the remaining ¼ cup. If it's sticky even after that, carefully mix in 1 tablespoon of flour at a time. Continue to knead the dough for 1-2 minutes after adding all of the flour.
Cover the bowl with the dough in it and let it sit in a warm area for about an hour. It should double in size. Make the cinnamon filling while waiting for the dough to rise.
Mix together the cinnamon, brown sugar, and butter with a spatula until it's fully combined.
Making the donuts
When the dough has doubled in size, sprinkle some flour onto a clean & dry countertop. Remove the dough from the bowl and fold it into thirds like a brochure. Flip it over, turn it 90 degrees, and fold it into thirds again.
Set the folded dough onto the lightly floured surface and roll it out into a 12x12 inch square. If it doesn't seem like it's rolling out, pick it up and carefully stretch it out with your hands. It may take a bit of work, but it will get there and it'll be pretty thin.
Carefully spread the cinnamon filling evenly across the entire surface of the dough (if it's too thick to spread, heat it in the microwave for a few seconds). I like to work it around with my hands, but you can use a spatula as well.
Fold the dough in half so that the cinnamon filling is enclosed and it's a 12x6 inch rectangle. Mark and cut the rectangle into 12 even strips (1x6 inch strips).
Put two strips on top of each other with the open sides on opposite ends. Twist them and then gently press the ends together so that it's a circle and place it in the donut pan hole. Repeat for all 6 donuts.
Let the donuts sit in the pan at room temperature while the oven preheats to 350°F. I recommend placing the donut pan on a sheet pan in the oven in case cinnamon filling bubbles over or you're using a flexible silicon donut pan.
Bake until the tops are lightly tanned and the cinnamon filling is bubbling (about 15-20 minutes). Make the cream cheese icing while they bake.
Allow the donuts to cool in the pan for about 15-30 minutes. Then, carefully remove them from the pan. Feel free to spread/drizzle the cream cheese icing on top while they cool in the pan or wait until after you remove them.
Cream Cheese Icing
Beat the cream cheese in a bowl until it's creamy. Stir in the butter, and then the powdered sugar. If you're making it a thin consistency, mix in the milk or cream (add more/less to your preferred thickness).
Store unfrosted donuts in an airtight container at room temperature for up to 1 week. Store frosted donuts in the refrigerator.