In a small bowl combine the yeast, warm water and the 1/4 teaspoon sugar - stir together and allow to sit for a few minutes to activate the yeast.
In a large mixing bowl add the 1/3 cup of sugar, warm milk, vanilla, eggs, oil, salt and 3 cups of the flour - once yeast mixture is activated add it to this bowl also.
Mix in the mixer until well combined and a sticky dough, change to the dough hook and gradually add the other cup of flour to make a firm, smooth, non-sticky, elastic dough.
Place the dough in a lightly greased bowl covered with a warm damp towel and let it rise until double in size, approximately 30-45 minutes.
While the dough is rising mix the cinnamon and sugar together in a flat bowl and set aside. Melt the 1/2 cup of butter in another bowl and set aside with the cinnamon sugar.
Punch the dough down and divide into small golf size balls and set aside covered while you heat your oil.
Heat about 4-5 inches of vegetable oil to 385° - I use my electric skillet so the temperature is easy to control and keep even. As you get ready to fry each one stretch the dough into the shape of a beaver tail, we are not looking for perfection here we are looking for deliciousness. I like mine to be a tad uneven - thin in some places for crispy bubbles and thicker in some places for more texture.
Once stretched carefully place the dough into the hot oil and allow to cook and brown for just a minute or less - watch carefully these do not take long! With tongs turn the beaver tail over to fry on the other side just till brown - remove from the oil and drain on a paper towel.
Allow to drain just a bit then brush the beaver tail with the melted butter on both sides and coat in the cinnamon sugar.