Preheat the oven to 375 degrees F (190 degrees C). Grease 12 large muffin cups, or line with paper liners.
In a large bowl, whisk together the eggs, applesauce, oil, brown sugar, vanilla and bananas. Combine the flour, baking soda and cinnamon; Stir into the banana mixture until moistened. Stir in the strawberries until evenly distributed. Spoon batter into muffin cups until completely filled.
Bake for 20 minutes in the preheated oven, or until the tops of the muffins spring back when pressed lightly. Cool before removing from the muffin tins.