1.Preheat oven to 350 degrees. Fit a large roasting pan with a rack. Place ducks on rack and prick duck skin all over with the tines of a fork, taking care not to pierce the flesh. Season ducks all over with sea salt; let stand 30 minutes.

2.Meanwhile, chop duck wings and neck; set aside. Heat oil in a large heavy-bottomed saucepan over medium-high heat. Add duck wings and necks and cook, stirring, until browned. Separate and peel garlic cloves from 1 head of garlic; reduce heat to low and add fennel and garlic cloves. Cook, stirring, until vegetables are softened. Add stock and cook until reduced by half, about 40 minutes. Strain into a small saucepan; skim fat from surface. Set sauce aside.

3.Halve remaining head of garlic and crush slightly. Divide lemon, thyme, and garlic evenly between duck cavities. Transfer duck to oven and roast, turning every 25 minutes, until duck begins to brown. Meanwhile, mix together vinegar and lemon juice. Once duck starts to brown, begin basting with vinegar mixture and cook until duck skin is dark brown and meat begins to come away from the breastbone, 2 to 3 hours total.

4.Remove from heat and let duck stand for 15 minutes. Meanwhile, reheat sauce over medium heat until warmed through and reduced to desired consistency. Carve duck and serve with sauce.