Mix together warm water, yeast, and 1 pinch white sugar in a large bowl; let sit until creamy, about 5 minutes.
Meanwhile, grease a separate large bowl and set it aside.
Stir warm milk, eggs, 1/3 cup vegetable oil, 1/3 cup white sugar, salt, and vanilla into yeast mixture until sugar is dissolved. Stir in 2 1/2 cups flour until no dry spots remain. Turn dough out onto a floured surface and knead in remaining flour, 1/2 cup at a time, until dough is no longer sticky. Knead until smooth and elastic, about 10 minutes.
Form dough into a ball and place it into the prepared bowl. Cover with a towel and let rise in a warm place until doubled in volume, about 1 hour.
Punch down dough and knead a few times to reshape it. Pinch off a piece of dough about the size of an egg, then roll into an oval shape, about 1/4-inch thick. Place on a dish towel, cover with another dish towel, and repeat with remaining dough.
Heat 4 inches oil in a deep fryer or large saucepan to 375 degrees F (190 degrees C).
Make cinnamon sugar: Combine white sugar and cinnamon in a shallow dish or pie plate; set aside.
Lower dough carefully into hot oil in batches and fry until golden brown, 1 to 2 minutes per side. Drain on a paper towel-lined plate. Gently press fried dough into cinnamon sugar while still warm; shake off excess sugar.