Whisk whole wheat flour, baking powder, baking soda, and salt in a bowl. Mix in brown sugar, breaking it up into pea-sized pieces. Beat eggs in a separate bowl and stir milk, applesauce, and vanilla extract into eggs. Make a well in the dry ingredients and pour in the milk mixture; stir until just combined. If batter is too thick, add more milk.
Grease a skillet or griddle with vegetable oil and place over medium-high heat. Scoop 2 tablespoons of batter onto the hot skillet and cook pancake until browned on the bottom and bubbles appear in the top, 2 to 3 minutes. Flip the pancake and cook until browned, 2 or 3 more minutes. Repeat with remaining batter.
