Mix warm water, yeast, and 1 pinch of sugar in a large bowl, and let stand until the yeast begins to foam and form a creamy layer, about 5 minutes. Stir in milk, 1/3 cup sugar, salt, vanilla extract, eggs, and 1/3 cup vegetable oil until the sugar has dissolved. Mix in about half the whole-wheat flour, and stir until the dough is too stiff to add more flour. Turn the dough out onto a floured surface, and knead in more flour until the dough is no longer sticky. Knead until smooth and elastic, about 10 minutes. Form the dough into a ball, place into an oiled bowl, and cover. Let rise until the dough has doubled in size, about 1 hour.
Punch down the dough, knead a few times to reshape it, and pinch off a piece of dough about the size of an egg. Roll the dough ball out into an oval shape about 1/4 inch thick. Place finished fried dough on a cloth and cover while you finish rolling out the pastries.
Heat oil in a deep-fryer or large, deep skillet to 375 degrees F (190 degrees C). Oil should be at least 4 inches deep. Combine 2 cups of sugar with cinnamon to taste in a shallow dish or pie plate, and set aside.
Gently lower the pastries, one at a time, into the hot oil, and fry 1 to 2 minutes per side until golden brown. Blot excess oil from fried pastries with paper towels; gently press each pastry into the cinnamon sugar while still warm. Shake off excess sugar.