In a large mixing bowl, combine the warm water, active dry yeast, and a pinch of the granulated sugar. Let the mixture sit for 5 to 10 minutes until it becomes frothy, indicating the yeast is active.

Whisk the warm milk, remaining 1/4 cup of sugar, melted butter, vanilla extract, and egg into the frothy yeast mixture. Stir continuously until the wet ingredients are well combined.

Gradually add the all-purpose flour and salt to the wet ingredients, mixing with a wooden spoon until a shaggy dough forms. Turn the dough out onto a lightly floured surface and knead for 5 to 7 minutes until it becomes smooth and elastic.

Place the kneaded dough into a lightly oiled bowl, cover it with a clean kitchen towel, and let it rise in a warm, draft-free place for about 1 hour, or until it has doubled in size.

Once risen, punch the dough down to release the air and divide it into 6 equal portions. Use a rolling pin or your hands to stretch and roll each piece into an oval shape about 1/4-inch thick, resembling a beaver's tail.

In a large, deep skillet or Dutch oven, heat the vegetable oil to 350°F (175°C). While the oil heats, mix the 1 cup of sugar and ground cinnamon in a wide, shallow dish for your coating.

Carefully lower the dough ovals into the hot oil, frying one or two at a time to avoid overcrowding. Cook for 1 to 2 minutes per side until they are puffed and a deep golden brown, then remove them with tongs and let the excess oil drip off for a few seconds.

Immediately toss the hot, fried pastries in the cinnamon-sugar mixture, pressing gently so the sweet coating adheres evenly to both sides. Serve warm immediately, optionally with a drizzle of chocolate hazelnut spread or a squeeze of fresh lemon juice.