Preheat the oven to 400°F (200°C). Squeeze the shredded parsnips firmly in a clean kitchen towel to remove moisture.
Toss the parsnips with melted butter, salt, and pepper. Press firmly into a 9-inch pie dish, building up the sides.
Bake the crust 15 to 20 minutes, until golden at the edges. Reduce the oven to 350°F (175°C).
Cook the ground beef and onion in a skillet over medium-high heat for 6 to 7 minutes until browned. Add garlic and thyme and cook 1 minute more. Season and drain excess fat.
Whisk together the eggs, milk, and heavy cream.
Scatter the beef mixture and cheese into the baked crust, then pour the egg mixture over the top.
Bake at 350°F (175°C) for 30 to 35 minutes, until just set.
Let the quiche rest 10 minutes before slicing.