Preheat your oven to 400°F. While it heats, prepare all your ingredients: dice the onion into 1/4-inch pieces, cut the broccoli into small florets (about 1/2-inch pieces), chop the fresh chives, and measure out the cheese, milk, and half and half. Having everything prepped and ready before you start cooking makes the process smooth and ensures nothing gets overcooked while you're still chopping.

Heat the olive oil in a large skillet over medium heat. Add the diced onion and cook for 5-7 minutes, stirring occasionally, until the onion becomes translucent and starts to soften. Add the broccoli florets along with 1/4 tsp salt and 1/4 tsp pepper, then continue cooking for another 5 minutes until the broccoli is just tender-crisp. You want the vegetables cooked through but still with a bit of firmness—they'll continue cooking in the oven, so don't overdo it here.

In a large bowl, whisk together the room-temperature eggs, half and half, milk, remaining 1/4 tsp salt, remaining 1/4 tsp pepper, and fresh chives. Whisk until the mixture is well combined and smooth. Using room-temperature eggs helps create a more tender, evenly cooked custard without hot spots.

Spread the sautéed onion and broccoli mixture from Step 2 evenly across the bottom of your pie pan or quiche dish. Sprinkle the shredded cheddar cheese over the vegetables in an even layer. Slowly pour the egg custard from Step 3 over the top, pouring gently so the cheese and vegetables stay somewhat distributed and the egg mixture fills in around them. Place the quiche in your preheated 400°F oven and bake for 30-35 minutes, until the center is just set but still slightly jiggly in the very middle—this ensures a creamy, tender texture rather than an overcooked, rubbery custard.

Remove the quiche from the oven and let it rest at room temperature for 5-10 minutes. This resting period allows the custard to set up a bit more and makes slicing cleaner. Slice into wedges and serve warm or at room temperature.