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, 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 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 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.