Mix the mayonnaise, yogurt, and mustard in a small mixing bowl; set aside.
Heat olive oil and butter in a skillet over medium; stir to combine.
Break open eggs and add to the skillet; season with salt and pepper. Cook until yolks are set as desired.
Meanwhile, slice the tomato and toast the bread. Spread mayo mixture onto each bread slice.
Place each finished egg onto a slice of bread. Add ham, cheese, lettuce and tomato; top with remaining bread slices.
Cut each sandwich in half and serve with bananas on the side.