To make the doughnuts
In the bowl of a stand mixer, whisk together yeast, warm water, and 1 Tbs. of the sugar. Let stand until mixture is foamy, about 5 minutes. Whisk in the remaining 2 Tbs. sugar, milk, vanilla extract, vanilla seeds, egg yolks, and butter. Fit in the mixer with paddle attachment and begin mixing on low speed.
Add flour and salt and mix for 3 minutes, scraping down the sides of the bowl and the paddle often to keep the dough moving.
Turn out dough onto a lightly floured work surface and knead by hand several times, dusting with flour as needed. Place dough in a large oiled bowl and cover with plastic wrap. Let rest in a warm place until doubled in bulk, about 1-1/2 to 2 hours. (Alternatively, you can let it rise in the refrigerator overnight, 8 to 12 hours.)
Cut out doughnuts with a 3-inch doughnut cutter, place on a parchment-lined baking sheet, and cover with a clean kitchen towel. Let rise for 30 to 45 minutes, or until doughnuts have doubled in size.
Pour 2-1/2 inches of vegetable oil into a 4- or 5-quart pot and heat it to 350°F. Fry doughnuts in batches of no more than 4 until they are deep golden brown. 2 minutes per side, turning only once (turning too often can result in greasy doughnuts). Transfer to paper towels to drain.
To make the glaze
For the glaze, whisk together the confectioners’ sugar, milk, salt, and vanilla seeds until smooth. Spoon glaze over warm doughnuts and serve.