Start by making anise water by placing 1¼ cups of hot water in a small bowl with anise. Let it steep for about 10 minutes, then strain and set aside 1 cup of anise-flavored water for the dough.
In a large bowl, mix the flour, baking powder, salt, and sugar.
Form a well in the center of the flour, and pour in melted butter, one cup of the anise water, and vanilla extract. Knead until a soft and smooth dough forms (about 2 minutes).
Cover the dough with a kitchen towel and let it rest while you clean up and flour your work surface.
Divide the dough into 12 small balls and roll each doughball out in a circular motion until as thin as possible without tearing. The thinner it is, the crispier it will be. Repeat with the remaining dough until all of them have been rolled out.
Heat about 2 inches of cooking oil in a heavy-bottomed or cast-iron skillet to around 350℉ (180℃).
Place the dough in the hot oil and fry until golden brown. It happens fast, so keep an eye on them. Transfer the fried bunuelos to a paper-towel-lined plate to drain excess oil.
Serve bunuelos warm or at room temperature with piloncillo syrup (instructions follow) or sprinkle with cinnamon sugar.