• Whisk sugar, ¾ cup salt, and 10 cups water in a large bowl until salt and sugar dissolve. Add lamb, cover, and chill for at least 12 hours and up to 2 days.
• Heat oil in a large heavy pot over medium-high heat. Add onions, garlic, rosemary sprig, thyme sprig, and bay leaf. Cook, stirring often, until onions are golden brown and soft, 10-15 minutes. Remove pot from heat and add wine, stirring and scraping up any browned bits from bottom of pan. Return pot to heat and simmer until wine is reduced by half, about 3 minutes.
• Remove lamb shoulder from brine; add to pot, along with stock. Bring to a boil. Reduce heat; simmer gently, partially covered and skimming fat occasionally, until lamb is fork-tender, 5-6 hours.
• Transfer lamb to a platter and remove bones. Cut or shred meat into bite-size pieces. If needed, boil cooking liquid in pot until reduced to 6 cups, 15-30 minutes. Season sauce with salt, if needed. Return lamb to pot.

Garlic Emulsion and Garnish

• Place eggs in a small saucepan and add water to cover by 1-inch. Bring to a boil and remove from heat. Cover and let sit for 3 minutes. Transfer eggs to a bowl of ice water and let cool completely. Peel.
• Combine eggs, garlic, 3 tablespoons lemon juice, and ½ teaspoons salt in a blender. With the motor running, gradually add oil, blending until a creamy sauce forms. Season garlic emulsion with salt and more lemon juice, if desired.
• Cook broccolini in a large pot of boiling salted water until crisp-tender, about 4 minutes. Drain; rinse under cold water.
• Divide lamb braise among bowls and garnish with broccoli, parsley, tarragon, and garlic emulsion.