Cut whole duck in half using kitchen shears, cutting along backbone. Remove innards and reserve. Place duck and innards on resting rack over sheet tray. Salt duck with kosher salt. Place duck, uncovered, in refrigerator to age minimum 8 hours or overnight.
Remove duck from refrigerator and fabricate breasts and legs. Reserve. Place duck carcass in large stock pot and cover with cold water. Let soak 1 hour. Strain. In large stock pot, heat vegetable oil over medium heat. Add onion, leek, celery, parsley with a dash of salt and saute 5-8 minutes until onions are translucent. Add duck carcass and saute another 5-10 minutes. Add cold water to cover and bring stock to a light simmer, careful not to boil over. Reduce heat to low. Use small ladle to remove all surface scum and grease as they float to the top. Let cook 6-8 hours – stock should reduce by roughly ½ in volume. Strain. Measure out final stock yield. Per 2 cups of broth, use 1 Tbsp of matcha powder. Place matcha powder in suitably sized bowl. Add hot broth, 1 ladle at a time, to matcha powder while whisking continuously to combine. Repeat for all of broth. Reserve.
When broth is ready, score breasts and place, skin side down, cold, in medium oven-safe saute pan. Turn up heat for saute pan with duck breasts to low and let render until duck breast skin is golden-brown and crisp. Flip breasts to skin side up and place in oven to cook 8-10 minutes, or to desired level of doneness. Let rest 5 minutes before slicing.
In medium stock pot, bring water to a rolling boil. Cook udon according to manufacturer’s specifications. Strain. Place udon into matcha broth and top with sliced duck breast. Garnish with scallion and sesame seeds.
PRO TIP: For a richer duck stock, bypass soaking the duck bones in cold water and instead roast them in a 400F oven for 15-20 minutes until golden brown.