In a small bowl, mix the white flour, baking powder, baking soda, and salt. Mix well to combine.
In a large bowl, mash the banana. To the mashed banana, add the melted butter, sugar, and vanilla extract. Mix well.
Add the milk and egg to the large bowl and mix again.
Pour the dry ingredients into the wet ingredients and mix until just combined. It's ok if there are a few lumps left in the batter. These will likely be from the mashed banana.
Cut the strawberry into small pieces and add them to the batter. Fold the strawberries into the batter, but try not to mash them.
Preheat a large skillet or a griddle over medium-high heat. Add a tablespoon of butter to the hot pan. Use a non-stick pan if you can which will prevent the pancakes from sticking.
Spoon the batter into the hot griddle or hot skillet. Let the batter cook for a couple of minutes. When the edges start to dry out and there are small bubbles in the center of the pancake, flip the pancake.
Cook the second side for 2-3 minutes. Continue with the rest of the batter until all the pancakes are made. Serve immediately.