1. Pour olive oil into a 12-inch ovenproof nonstick frying pan (with at least 2-in.-tall sides) over medium heat. When hot, add carrots, celery, shallots, and garlic; stir often until vegetables are slightly limp and beginning to brown, about 7 minutes. Add wine, orange juice, 5 sprigs thyme, and bay leaves; increase heat to high and boil, stirring occasionally, until liquid is reduced by about half and is slightly syrupy, about 15 minutes. Pour through a fine strainer into a glass measure; you should have about 1 cup. Discard vegetables. Wipe pan dry.
2. Rinse duck breasts and pat dry. Score the skin of each in a crosshatch pattern, making cuts about an inch apart, through the skin into the layer of fat. Sprinkle lightly all over with salt and pepper. Set same frying pan over medium-high heat; when hot, lay duck breasts, skin side down, in pan. Cook until beginning to brown on the bottom, 1 1/2 to 2 minutes. Spoon out and discard any fat in pan. Turn breasts over and sprinkle evenly with thyme leaves.
3. Transfer pan with duck to a 425° oven. Roast until well browned on the surface but still slightly pink in center of thickest part (cut to test), 20 to 25 minutes. Transfer duck to a rimmed cutting board and let rest in a warm place for 5 minutes. Skim off and discard fat from pan juices.
4. Set pan over medium-high heat and add orange-wine reduction and cream. Stir often, scraping up browned bits from bottom of pan, until sauce is boiling and coats the back of a spoon in a thin layer, 2 to 3 minutes. Season to taste with salt and pepper.
5. Divide arugula among six plates. Slice duck breasts across the grain and fan equally over arugula. Drizzle a little warm orange-wine sauce over duck and greens. Pour remaining into a small pitcher and pass, for guests to add to taste.