In the bowl of a stand mixer fitted with the beater attachment, combine the yeast, milk, salt, and sugar. Set aside and allow the yeast to bubble or see some movement. Note: you don't actually have to let instant yeast "activate" but I like to do it anyway. Let's me know I'm set up for success!
Once the yeast is "activated", add the eggs, butter, and vanilla. The batter will look really strange, it will be mostly liquid with tiny little balls of butter. But that's okay! You just want to break things down and combine as much as possible before you add your flour.
Add your flour and continue to mix on medium speed until a soft dough has formed. Make sure that you fluff and scoop your flour when measuring in your cups. You don't want to pack the flour into the measuring cups, or your dough will be too stiff. The dough should be slightly sticky but very workable.
At this stage, you can either remove the dough from the KitchenAid and knead the dough by hand, or you can switch your beater for your dough hook and finish kneading with the mixer.
If you are kneading by hand, I recommend using the folding method, where you stretch the dough away from yourself and then fold back over. Then turn the dough 90 degrees and repeat. Either way you choose to knead your dough, continue kneading until a smooth ball of dough has formed.
Cover the bowl of dough with a kitchen towel and set aside in a warm place to double in size. This will depend on several factors, including the temperature of your kitchen, the temperature of the milk you used, how you chose to knead your dough, etc. but this generally takes about an hour for me. But I keep our house pretty warm! So focus more on the dough doubling in size, and less on the timing. I like to take a picture of the dough before I set it aside so I can compare it to what it looks like later to make sure it has truly doubled.
When you poke the dough and the imprint of your finger mostly remains, but springs back slightly, the dough is ready to be rolled. If the fingerprint fills in quickly and all of the way, let it continue rising. If the fingerprint doesn't spring back at all, the dough is overproofed. You can still continue on with the recipe, but the dough may be a little tougher than you would want.