Combine shallot, sugar, salt, pepper, garlic, and thyme in a small bowl. Rub all over duck legs, thighs, and wings.

Arrange duck parts skin side-up in a dish and cover with plastic wrap. Refrigerate until the seasoning is well absorbed, 24 to 48 hours.

Rinse off the seasoning and pat the duck dry. Arrange duck in a single layer in a 9x13-inch baking pan. Let duck stand until it reaches room temperature, about 30 minutes.

Preheat oven to 225 degrees F (110 degrees C).

Melt duck fat in a saucepan over medium heat until bubbles begin to form, 6 to 8 minutes. Pour duck fat over the room-temperature duck.

Bake duck in the preheated oven until tender and the juices run clear, 2 to 3 hours. An instant-read thermometer inserted near the bone should read 135 degrees F (57 degrees C). Pour off the duck fat.

Brush a grill pan with some of the used duck fat and heat over medium-high heat until smoking. Add baked duck; cook in batches until skin is crispy, about 90 seconds per side.