Preheat the oven to 375 degrees F (190 degrees C). Grease an 8x8-inch baking dish.

Heat vegetable oil in a skillet over medium heat. Add onion; cook and stir until softened and translucent, about 5 minutes.

Reduce heat to medium-low; continue cooking and stirring onion until very tender and dark brown, 15 to 20 minutes more. Set aside.

Beat mayonnaise, eggs, flour, salt, and pepper together in a large bowl. Stir in broccoli and onion. Pour into the prepared baking dish.

Bake in the preheated oven until broccoli is tender, about 90 minutes.