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Set the oven rack in the middle position. Preheat to 350°F (177°C).
Grease a 9-inch pie dish with baking spray or brush with olive oil.
Heat a large skillet over medium heat. Once hot, add the olive oil. Add the onions and saute until translucent, about 2 to 3 minutes. Add the broccoli and sprinkle with ½ teaspoon salt, saute until lightly browned and tender, about 4 minutes. Add the garlic, saute until fragrant, 30 seconds. Turn the heat off.
Evenly sprinkle half of the cheddar cheese into the greased pie dish. Top with the sauteed broccoli mixture, followed by the remaining cheese.
In a medium bowl, whisk together the eggs, heavy cream, whole milk, ½ teaspoon salt, pepper, and nutmeg. Slowly pour the egg mixture into the pie dish, allowing it to settle around the ingredients and to the bottom. Use a fork to move the ingredients to help the custard settle. Sprinkle the surface with some black pepper.
Bake the quiche on top of a sheet pan in the middle of the oven until the surface is lightly golden brown, the center is just set, and the edges slightly puffy, about 30 to 35 minutes. The temperature should be about 190 to 195ºF (88 to 91ºC) in the center and about 195 to 200ºF (91 to 93ºC) on the edges.
Cool on a wire rack for 5 to 10 minutes before serving. Slice or cool completely, cover, and refrigerate to serve the next day.