In the bowl of a stand mixer fitted with the hook attachment, mix the flour, salt, and nutmeg. Set aside. Whisk the yeast and the sugar in a small bowl. Set aside.
Heat the milk to 110F (if you don’t have a thermometer, I usually microwave it for 30 seconds on high, which gets me to about 110-120F).
While whisking constantly, slowly stream in the sugar/yeast mixture into the heated milk. If you are using active dry yeast, allow the mixture to sit for 15 minutes. The yeast will activate in this time. If using instant yeast, you don’t have to wait for the yeast to activate, simply move on to the next step.
Mix the eggs and vanilla extract into the milk/yeast mixture. Whisk well to combine.
Pour the milk/yeast/egg mixture into the flour and mix on low-medium speed for about 3 minutes, or until it forms a ball around the hook.
With the mixer on medium speed, add the butter a few pieces at a time, allowing them to fully incorporate before adding more. While you’re incorporating the butter, don’t be quick to add more flour if the dough looks too wet. Let it knead for a little before adding anymore butter and it will come together & look less sticky, then continue to add the butter. Adding too much flour preemptively will result in a dry bread. Once all of the butter is fully incorporated, turn the mixer up to medium-high speed and knead for an additional 8-12 minutes (mine usually takes about 10 minutes).
To check if the dough has kneaded enough, rip off a small piece and spread it gently in between your fingers. If it stretches easily without ripping and you can see light passing through it, it is finished and ready to proof.
Transfer the dough to a lightly greased bowl, cover with a towel, and let it rise in a warm environment for about 1 hour, or until doubled in size. Alternatively, cover the bowl with plastic wrap and let it rise in the fridge overnight.
While your dough is proofing, cut out thirteen 4 inch parchment paper squares and arrange them on a baking sheet. If you are proofing the dough overnight, simply just do this before you begin.
Once the dough has doubled in size or it’s the next day, turn it out onto a non-floured surface - this will make it easier to shape the dough. Divide it into 13 equal pieces (each should roughly weigh about 70 grams and you should have a very small amount of dough left over). I like to use a bench scraper to cut the dough.
To shape, flatten each piece on your surface, then take all of the sides and gently fold them upwards and inwards, pressing them into the center. Turn the dough over and roll into a smooth ball. Place each piece of dough on a parchment paper square on your pan. Repeat this with your remaining dough, then cover and let rise for another 30ish minutes.
In this time, begin to heat up your oil. You can fry these in a large pan, cast iron dutch oven, or a deep fryer. Use a thermometer to maintain a temperature of around 360F.
Press your finger into a roll - if it bounces back immediately, it needs more rest. If it doesn’t bounce back at all, it is over-proofed and should be fried immediately. If it bounces back slowly, it’s ready.
Carefully lower 1-2 donuts at a time into the oil. If the oil is bubbling rapidly when you drop a donut in, it is way too hot. The edges will get too dark before the inside is cooked. If the oil doesn’t bubble at all, it is not hot enough. The donuts will take too long to fry and they will be greasy. You want the oil to bubble gently. This is why a thermometer is important!
Fry the donuts for 60-90 seconds on each side. Carefully remove from the pan and place onto a baking sheet lined with paper towels to soak up the excess oil. Then move to a cooling rack. Repeat with the rest of your donuts. While you are frying your first few donuts, the other donuts can just stay on the pan with the towel over them - the time they are waiting to be baked won’t really affect the proofing process too much if you are working efficiently.
Whisk the granulated sugar & cinnamon in a shallow bowl. Once cool enough to handle, toss the donuts in the cinnamon sugar. Set aside to continue cooling.
Remove the caramel from the fridge and empty it into a bowl. Whip the caramel using a hand mixer on medium speed until lighter in color and whipped - it won’t get super thick, plus it’s going to get all melty in the donuts, so don’t stress if you think yours doesn’t look whipped enough. Mostly, I do this for the color.
Filling the donuts: I like to use a cake dowel to make a hole in the side of the donut - gently press into the donut and wiggle your dowel around to make space for the filling to go. You can also do this with a chopstick, toothpick, etc. If you don’t have any one of those things, you can try a knife (but be careful!) or use the piping tip to wiggle your way into the donut. Fill the donuts using a piping bag for ease. Enjoy!