In a large mixing bowl, whisk together 2 cups all-purpose flour, 1/2 cup granulated sugar, 2 teaspoons baking powder, and 1/4 teaspoon salt. In another bowl, beat the 2 large eggs, 1/2 cup milk melted 1/4 cup butter, and 1 teaspoon vanilla extract until smooth. Gradually combine the wet and dry ingredients to form a soft dough.
Pour oil into a deep fryer or large pot to a depth of about 2 inches. Heat the oil to 350°F (175°C). Use a thermometer for best results.
Lightly flour your work surface. Roll out the dough to about ½ inch thickness. Use a donut cutter or two round cutters (one large, one small) to cut out donut shapes and holes.
Carefully place the donuts into the hot oil, a few at a time to avoid overcrowding. Fry for 1–2 minutes per side, until golden brown. Remove with a slotted spoon and place on paper towels to drain excess oil.
While still warm, dip the donuts in your favorite glaze or sprinkle with powdered sugar. Let them cool slightly, then serve fresh and enjoy!