Using the largest holes on a box grater, shred potatoes. Place in a large bowl and toss with 1 tsp. salt and 1/2 tsp. pepper. Transfer potatoes to a clean dish towel, gather together the ends of towel and thoroughly wring out excess liquid over the sink. Transfer potatoes to a bowl and set aside.
Heat oil and 2 Tbsp. butter in a 10" (top diameter) cast-iron skillet over medium-high until butter is melted. Add potatoes and immediately start forming into a crust by pushing potatoes flat against bottom and sides of pan with a 1/2-cup dry measuring cup. Continue cooking, pressing potatoes up sides of pan if they start shrinking, until potatoes are bound together and the bottom of the crust is starting to brown, about 10 minutes. Remove pan from heat and set aside.
Meanwhile, melt 1 Tbsp. butter in a small skillet over medium heat. Add shallots and sauté until translucent, about 5 minutes. Set aside.
In a large bowl, whisk eggs, half-and-half, salt, pepper, mustard powder and nutmeg. Set aside.
Sprinkle cheese and sautéed shallots evenly over bottom of the potato crust, then pour in egg mixture. Arrange asparagus
(and chives, if using) decoratively on top.
Bake until quiche is set and crust is well browned, 30-35 minutes. Let cool at least 10-15 minutes, or to room temperature before cutting into wedges and serving from the pan.