Preheat oven to 375°; put broccoli in a steamer basket and steam over boiling water until tender, almost 4 minutes; remove the steamer basket from the saucepan; set aside.
Combine the oil, garlic, and red pepper flakes in a big skillet; cook, stirring, over med-low heat until the garlic begins to sizzle.
Add in the broccoli and cook for 1 minute more.
Remove from heat; stir in the onion and sprinkle with the salt and a grinding of pepper.
In a big bowl, whisk the eggs until light; stir in the half-and-half.
Spoon the broccoli mixture evenly into the pie shell; sprinkle with the cheese and pour in the egg mixture.
Bake the quiche until puffed and golden brown, about 35 minutes; (cover pie crust edges to prevent overbrowning).
Serve warm or at room temperature, cut into wedges.