1/2 cup warm water
1 tablespoon active dry yeast (from two 1/4-ounce envelopes)
4 1/2 cups all-purpose flour, plus more if needed
4 large eggs
1 1/4 cups sugar
2 teaspoons coarse salt
1 stick unsalted butter, softened, plus 1/2 stick, melted, and more for dish
2 1/2 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
make cinnamon-bun bites:
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 paste. Let rise in a warm place until yeast mixture has risen and fallen, about 30 minutes.
TIP: If your house is chilly like mine, turn on your oven for a couple of minuets on the lowest temperature, then turn off oven and place the bowl in the oven as sort of a proofing boxing for the dough.
Add eggs, 1/4 cup sugar, salt, 1/2 teaspoon of cinnamon, a pinch of nutmeg, and the remaining flour to yeast mixture. Knead dough, using dough hook attachment on mixer, until smooth, 4 to 5 minutes.
Add 1 stick butter to the dough mixture and knead with the stand mixer 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, pressing out air bubbles.
Lightly butter a 9-inch pie plate or 9-inch square glass baking dish. Place melted butter in a bowl. Stir together remaining 1 cup sugar and the cinnamon in another bowl.
Roll dough into 1-inch balls (you should have about 45). Working with 1 at a time, dip balls into melted butter, then roll in cinnamon sugar 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. Unwrap balls and place the pie plate on a baking sheet. Bake until golden and firm, and a wooden skewer inserted into the center comes out clean, 50 to 55 minutes. Let bites cool in dish 15 minutes, then drizzle with glaze.
make vanilla glaze:
Stir together sugar and salt in a small bowl.
Whisk together butter, milk, vanilla, and a pinch of cinnamon and nutmeg. 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.