Combine flours, baking powder, sugar, salt in a large bowl. Mix with a fork. In another bowl, whisk together milk, egg, melted butter. Stir in cheese and bacon.
Pour milk and cheese mixture into flour mixture, stir until just combined. Some lumps will remain.
Heat a pan or griddle over medium-high heat. Coat it lightly with oil. When griddle is hot, pour batter onto griddle in even circles. Cook until bubbles form, start to break in batter. Flip, cook until done.