In a saucepan, combine the butter, water, salt, and 1 1/2 teaspoons of the granulated sugar; bring to a boil. Remove from the heat. Add the flour and stir with a wooden spoon until combined. Return the saucepan to high heat and cook the dough, stirring until very smooth, 1 minute. Transfer the dough to a bowl. Using a handheld electric mixer, beat in the vanilla, then beat in the eggs, one at a time.

In a saucepan, heat the oil to 375 degrees. Line a baking sheet with paper towels. Scoop 10 scant tablespoons of the dough into the oil and fry, turning, until golden and puffed, 6 minutes. Transfer the beignets to the baking sheet and poke a small hole in the side of each to release steam. Repeat with the remaining dough.

Pour the 1 1/2 cups of granulated sugar into a bowl. Toss the hot beignets in the sugar; transfer to a platter. Dust with confectioners' sugar and serve.