In the bowl of a stand mixer, whisk together the yeast, 1 tbsp sugar and the warm milk. Cover and let bloom for 5 mins. It should look foamy.
In a separate bowl, combine flour, salt and remaining sugar.
To the foamy yeast mixture, add ½ or about 2 tbsp of the beaten egg (reserve the rest to brush the rolls before baking) and the flour mixture. With the dough hook attachment, mix on medium low speed until the dough is shaggy, about 2-3 minutes. While the mixer is running, add the butter one piece at a time, making sure it’s incorporated before adding the next. The dough will look moist and slippery until it’s all incorporated and fully kneaded. Do not add any extra flour here! Increase the speed to medium and knead until the dough comes together smoothly and pulls away from the bowl, about 8 minutes. See Note 2.
Remove the dough from the bowl. Lightly grease the same bowl or a different one, and place the dough it it. Cover with plastic wrap or a damp kitchen towel, and let it rise in a warm area until it’s doubled in size and puffy, about 1 ½ hours. I place mine in the oven with the ‘proof’ function and the oven light on.
Meanwhile, grease and 8 x 8-inch baking pan with vegan butter.