Preheat oven to 350 degrees. Melt 1 tablespoon butter in a large skillet over medium. Add onion and chard leaves; season with salt and pepper. Cook, stirring occasionally, until chard wilts and becomes tender, 6 to 8 minutes. Transfer to a plate; let cool completely. Meanwhile, blanch chard stems in a pot of generously salted water until crisp-tender and more vibrant in color, 1 to 2 minutes. Drain; transfer to another plate and let cool completely. Melt remaining 7 tablespoons butter in a saucepan. Brush a 10-inch deep-dish pie plate or straight-sided skillet with melted butter. Place 1 sheet of phyllo in pie plate; brush with more butter. Continue layering phyllo in overlapping concentric circles, brushing each layer with more butter. Transfer pie plate to rimmed baking sheet.

Whisk eggs in a large bowl. Whisk in milk, cream, and Dijon; season with salt and pepper. Spread chard-leaves mixture in phyllo crust; pour egg mixture over. Sprinkle nutmeg over top.

Bake until eggs begin to set at edges, about 25 minutes. Scatter chard stems evenly over top. Continue baking until eggs are set in center and puffed slightly and phyllo is golden brown, 40 to 50 minutes more. Let cool on a wire rack 15 minutes. Stir together sour cream and harissa; serve with quiche.