Preheat the oven to 350°F (180ºC), or 160°C for a fan oven, and adjust the oven rack to the bottom of the oven.
In a large bowl, whisk together the eggs, milk, salt and pepper. Add the Bisquick baking mix, and whisk until completely smooth. Set aside.
In a medium saute pan, heat oil over medium-high heat. Add the onion, and cook until soft and translucent, about 2 minutes.
Add the broccoli and cook for one more minute, then add garlic and cook for 30 seconds or until fragrant. Remove from heat.
Spray a 9 or 10-inch pie dish with nonstick spray, then add the vegetables and sprinkle the cheese on top.
Give the Bisquick mixture a quick whisk, and pour it all over the vegetables and cheese.
Bake in the preheated oven for 35 minutes, or until the quiche puffs up and the top is golden brown.
Remove from the oven, allow to set for at least 10 minutes before slicing and serving. The quiche will deflate as it cools down.