In a mixing bowl, whisk together the dry ingredients: all-purpose flour, granulated sugar, baking powder, and salt.
In another bowl, beat the eggs lightly, then add milk (or buttermilk), melted butter (or vegetable oil), and vanilla extract (or other flavorings). Mix until well combined.
Gradually add the wet ingredients to the dry ingredients, stirring until just combined. Be careful not to overmix, as this can result in tough donuts.
Turn the dough out onto a floured surface. Roll it out to about 1/2-inch thickness.
Use a donut cutter or cookie cutter to cut out donut shapes. You can also use a knife to cut out donut shapes if you don't have a cutter.
In a deep fryer or heavy-bottomed pot, heat vegetable oil or canola oil to 350°F (175°C). Use enough oil to submerge the donuts completely.
Carefully place the donuts into the hot oil, a few at a time, depending on the size of your fryer or pot. Be cautious not to overcrowd the oil.
Fry each side until golden brown, about 1-2 minutes per side, using a slotted spoon or spider strainer to flip the donuts and remove them from the oil.
Transfer the fried donuts to a wire rack lined with paper towels to drain excess oil.
Allow the donuts to cool slightly before adding toppings or glaze. Enjoy!