1/2 cup warm water
1 tablespoon active dry yeast
4 1/2 cups all-purpose flour, plus more if needed
4 large eggs
1 1/4 cups sugar
2 teaspoons coarse salt
1 teaspoon vanilla
1 stick unsalted butter, softened, plus 1/2 stick, melted, and more for coating the dish
2 tablespoons ground cinnamon
1 cup confectioners’ sugar, sifted
1/2 teaspoon coarse salt
2 tablespoons unsalted butter, melted
2 tablespoons whole milk
1 teaspoon pure vanilla extract
Pour water into the bowl of a stand mixer. Sprinkle yeast evenly over water, and let stand until foamy, 5 to 7 minutes. Whisk until smooth, then whisk in 1/2 cup flour to form a loose paste. Let rise in a warm place until yeast mixture has risen and fallen, about 30 minutes.
Add eggs, 1/4 cup sugar, salt, vanilla, and remaining flour to yeast mixture. Knead dough, using dough hook attachment on mixer, until smooth, 4 to 5 minutes.
Add 1 stick butter, and knead until dough is smooth and elastic, about 10 minutes (add a little more flour if the dough is too sticky). Turn out dough onto an unfloured work surface, and knead by hand until very smooth, about 3 minutes. Transfer dough to another bowl, and let rise in a warm place, covered with a damp cloth or plastic wrap, until doubled, 1 1/2 to 2 hours. Punch down dough; press out air bubbles.
Lightly butter a 9-inch pie plate. Place melted butter in a bowl. In another bowl,stir together remaining 1 cup sugar and the cinnamon.
Roll dough into 1-inch balls (I ended up with about 55). Working with 1 at a time, dip balls into melted butter, then roll in the cinnamon-sugar mixture and place in pie plate. Stack balls to form a dome. Sprinkle with any remaining cinnamon-sugar, and drizzle with any leftover butter. Cover with plastic wrap, and let balls rise by half, about 1 hour.
Preheat oven to 350 degrees F. Unwrap balls, and place pie plate on a baking sheet. Bake until golden and firm (a wooden skewer inserted into the center comes out clean), 50 to 55 minutes. Let bites cool in dish for 15 minutes.
In the meantime, let’s make the glaze. First, stir together sugar and salt in a small bowl.
Then, whisk together butter, milk, and vanilla, and stir into sugar until mixture forms a smooth paste. Keep glaze at room temperature until ready to drizzle over cinnamon-bun bites. Before using, whisk until smooth.