Preheat the oven to 350°F.
Bring a large pot of water to a boil. Add the broccoli florets and blanch for 5 minutes. Remove from the water, drain, and pat dry.
Heat a drizzle of olive oil in a frying pan on MED-HIGH. Add the onions. Sauté for 5 minutes until they are fragrant. Set aside.
In a large bowl, add the eggs. Add the cooked onion, mayonnaise, chicken soup, and 1 ½ cups of the grated cheddar. Mix together.