Peel the banana and in a medium bowl, use a dinner fork to thoroughly mash the banana.
Add the vanilla essence.
Pour the eggs over the banana and stir until completely combined. The batter will be very loose and liquidy, more like whisked eggs than regular pancake batter.
Heat a pan, over medium heat. Melt a little butter or warm the vegetable oil in the pan.
Drop about 2 tablespoons of batter into the pan. It should sizzle immediately — if not, turn up the heat slightly. Repeat with dropping more batter into the pan, leaving at least an inch or two between pancakes.
Cook the pancakes until the bottoms look browned and golden when you lift a corner, about 1 minute.
Gently work a thin, flat spatula about halfway under the pancake, then lift until the unsupported half of the pancake is just barely lifted off the skillet. Flip the pancake.
These pancakes are best when eaten fresh off the griddle and still warm. Serve with maple syrup, honey, jam, or fresh fruits.