Preheat oven to 350° F (177° C).
Heat broccoli florets in a small sauté pan over medium heat and dry sauté without any oil for 5-10 minutes until cooked and starting to lightly brown.
While broccoli is cooking, crack eggs into a medium sized bowl. Add milk, mustard, salt, and pepper. Whisk with a wire whisk until bubbly and mixed thoroughly.
Spread broccoli, ham, and cheese out in a silicone 9" round baking dish or a buttered 9" pie tin (make sure it is buttered thickly to keep from sticking). If using a silicone baking dish, make sure to put it onto a metal sheet pan to allow it to be easily put into and pulled from the oven.
Pour egg mixture evenly into baking dish. Place pan in oven and cook for 35-40 minutes until quiche is cooked through, starting to lightly brown, but still jiggly in the center.
Remove quiche from oven and let cool for 15 minutes before cutting and serving. Keep leftovers for 3-4 days in the refrigerator, reheating using the microwave.