Make the crust: Pulse flour, salt, and pepper in a food processor until combined, about 4 pulses. Scatter butter pieces over flour mixture; pulse in 1-second bursts until butter is the size of small peas, 9 to 10 pulses. Drizzle ice water into mixture, 1 tablespoon at a time, pulsing until mixture resembles coarse meal. Turn out onto a work surface; knead just until dough comes together. Using your hands, flatten into a 1-inch-thick rectangle. Wrap in plastic wrap, and chill 1 hour.

Line bottom of an 18-x 13-inch rimmed baking sheet with parchment paper. Unwrap dough, and roll out on a floured work surface into a 20-x 15-inch rectangle. Gently roll dough around rolling pin, and unroll onto prepared pan. Press dough into corners and up sides of baking sheet. Trim excess dough to about 1/4 inch above rim of pan. Crimp edges of dough as desired. Freeze 45 minutes.

Preheat oven to 400°F with rack in upper third of oven. Line crust with 2 large sheets of parchment paper; top with dried beans or pie weights to cover. Bake in preheated oven until bottom is set and edges are light golden brown, 35 to 40 minutes. Remove weights and parchment. Return crust to oven; bake at 400°F until crisp and browned, 15 to 20 minutes. Gently press crust down if needed. Let cool completely.

Make the filling: Heat oil in a medium skillet over medium. Add sausage; cook, breaking into medium-size pieces with a wooden spoon, until just cooked through, about 6 minutes. Remove from heat.

Whisk eggs in a large bowl until smooth. Whisk in half-and-half, salt, and pepper. Sprinkle cheese, sausage, and about half of onion rings over crust. (Reserve remaining onion for another use.) Gently pour egg mixture over crust; sprinkle with thyme.

Bake at 400°F until slightly puffed and just set, 30 to 35 minutes. Remove from oven; let stand 10 minutes. Top with arugula just before serving.