Put a gallon of water in a large pot. Add 1 tsp of salt and 6 eggs to the water.
Bring the water to a boil, and then lower to a simmer. Simmer for 5 minutes.
In the meantime, put ice and water in a large bowl. (This will be used to shock eggs and broccoli.)
Remove eggs from the pot using a strainer and transfer to the ice water. Bring the pot of water back to a boil, and add broccoli florets to the pot.
Boil for 2 minutes. Strain florets and add them to the ice bath with eggs. As soon as broccoli becomes cold, drain the ice bath, and peel the eggs. If broccoli is very wet, blot them with a towel before dressing.
To make kochujang-mayo dressing: place mayo, kochujang, sesame oil, 1/2 tsp salt, minced garlic, and lemon juice to a large bowl.
Take two eggs and gently cut in half without cutting through the yolk. (Reserve the two egg yolks.) Chop up the egg whites plus two whole eggs and add to the dressing. Coat the broccoli with the dressing.
Place dressed broccoli on a serving plate. (If making ahead. At this point plastic wrap plate and refrigerate.) Using a strainer or microplane, grate the cooked egg yolks over the salad. Cut up the remaining two eggs in half . Use the halved eggs for decoration.