Combine the flour, baking powder, and salt and pepper in a mixing bowl. Mix well.
In another bowl, combine the milk, egg, and olive oil. Whisk it into the dry ingredients until well incorporated; do not overmix. Set the batter aside for 30 minutes.
Heat the corn or peanut oil to 375*F. in a deep heavy skillet.
Using tongs, dip the artichoke hearts in the batter and then gently drop them in the hot oil. Fry a few at a time until golden brown, just a few seconds. Drain on paper towels, repeat with the remaining artichokes. Serve hot, with the bearnaise sauce.
Note: If you don't have a deep fat thermometer, drip a 1" cube of bread in the oil. If it turns golden brown within 1 minute, the oil is hot enough. Remove the bread and proceed with the recipe.