Step 1Salt the duck legs with about one-half teaspoon salt for each leg. Cover and refrigerate overnight.
Step 2Rinse the duck legs with cold water and pat them dry.
Step 3Heat the oven to 350 degrees. In a large ovenproof pot, bring to a simmer over medium heat enough duck fat to cover the legs. Once the fat has come to a simmer, place the legs in the pot. Cover and cook in the oven for about 2 hours.
Step 4Allow the duck legs to cool in the fat. Using tongs, remove the duck legs from the pan. Remove the skin and pull the meat from the legs.
Step 1In a large bowl, mix the duck confit with the ground duck breasts. Mix in the red onion, chives, freshly ground white pepper, olive oil, egg yolk and salt.
Step 2Form four patties, about 7 to 8 ounces each. Heat the oven to 375 degrees.
Step 3In a very hot saute pan, sear the patties for about 1 minute on each side and finish in the oven for 4 to 6 minutes, until medium rare.
Step 4Wipe clean the saute pan and grill the buns until they are just crisp.
Step 5Place each burger on a crisp bun, and top with a slice of heirloom tomato and a couple leaves of butter lettuce.
Step 6In the saute pan, heat the butter over medium-high heat for a few minutes. Fry the eggs -- sunny side up -- for about 1 to 1 1/2 minutes, until the egg whites have solidified and set.
Step 7Place an egg on top of each burger. Serve open face with a drizzle of Dijon creme fraiche.