Whisk together 3 cups flour, 6 tablespoons sugar, the yeast, and salt, and set aside.
Combine milk and eggs in the bowl of a mixer fitted with a dough hook. Add flour mixture, and mix on medium speed 2 to 3 minutes, or until dough comes together and forms a ball.
Add the soft butter, slowly, one piece at a time. Continue mixing for about 3 minutes, and add remaining flour 1 tablespoon at a time, until dough forms a soft ball (you may not need the full 3/4 cup).
Place dough in a lightly oiled bowl, and cover with plastic wrap. Place in a warm place, and let rise until doubled in size, 2 to 2 1/2 hours.
Turn out dough onto lightly floured surface, and pinch off 1 1/2-ounce pieces, about 1 inch in diameter. Place on a lightly floured cookie sheet, cover with plastic wrap, and let rise in a warm place for 30 to 45 minutes. Meanwhile, mix cinnamon and remaining 1 tablespoon sugar in a wide bowl.
Heat oil to 365 degrees, and drop donut holes into oil in batches. Cook until golden brown, about 6 minutes. Transfer to paper towels to drain, and toss with cinnamon sugar while still warm. Let sit 5 minutes before serving; let cool completely before packaging in glassine bags as favors.