Place duck on a baking sheet. Rub with finely chopped thyme, salt, and ginger. Cover with plastic wrap; chill 4 hours or up to 8.
Brush excess spice mixture from duck and transfer to an 8-qt. saucepan. Add thyme sprigs, stock, peppercorns, garlic, bay leaves, and ginger; bring to a boil. Reduce heat to medium-low; cook, covered slightly, until meat is very tender, 1 ½–2 hours. Remove from heat and let cool in the pan until room temperature; cover and chill overnight.
Next day, uncover and scrape solidified fat from top of pan; set aside. Pull duck from pan and discard skin, bones, and gristle; finely shred meat. Strain and reserve ¼ cup stock; save remaining stock for future use. Transfer meat to a bowl and add reserved stock, plus armagnac, parsley, orange zest, cloves, salt, and pepper; stir to combine. Tightly pack rillettes into a 1-qt. baking dish or five 6-oz. ramekins. Melt reserved fat plus 1 cup duck fat in a 1-qt. saucepan over medium heat; pour over rillettes; cover with plastic wrap and chill until fat is solid. To serve, push fat aside and spread rillettes on toast points; garnish with cornichons or pickled vegetables, if you like. To store, redistribute fat over rillettes and cover with plastic wrap; keep, refrigerated, up to 2 weeks.
