Combine flour, baking powder, and salt in a large mixing bowl and whisk together. Gradually whisk in the milk, then the egg and vanilla, and finally the sugar and melted butter.
Heat the oil in a large, heavy pot over high heat to 350 degrees F. Pour as much batter as fits into a funnel, holding your finger at the base of the funnel to dam it. Bring the funnel over the top of the hot oil, release your finger to begin the stream of batter, and move the funnel in a circular motion to create a spiral shape. Use about 1/2 cup batter per funnel cake, or more or less to your liking. Fry 2 to 3 minutes until golden brown and slightly puffed, turning over when 1 side is golden.
Remove from oil and place on a plate lined with 2 layers of paper towels. Immediately top with a good scattering of powdered sugar. Repeat until no batter remains.