Bring the lamb to room temperature. Pat the lamb shoulder dry with paper towels, then let it sit out at room temperature for at least 30 minutes before cooking.
Prepare the lamb. Preheat an oven to 425°F (220°C). Place red onion quarters, halved shallots, halved garlic, 2 halved lemons, fresh oregano, and fresh rosemary in a roasting pan or large Dutch oven. Place the lamb skin-side down on the herbs, then season with half of the olive oil, kosher salt, and black pepper. Flip the lamb over so that it’s skin-side up, then drizzle with the remaining olive oil and season with remaining salt and pepper.
Cook the lamb. Transfer to the preheated oven and cook for 30 minutes, then carefully remove. Reduce the oven temperature to 350°F (175°C). Add 1 cup dry white wine and 1½ cups low-sodium chicken broth, cover tightly with aluminum foil, then return to the oven. Cook for 3½-4 hours. Remove the aluminum foil, then cook for a final 30 minutes or until the lamb is absolutely fork-tender.
Allow the meat to rest. Loosely tent the lamb with foil, then rest for 20-30 minutes before serving. Serve with the herb sauce spooned on top (instructions below) and a pinch of flaky sea salt. If you want, you can enjoy with the cooked onions and shallots!
To make the herb sauce (in a personal blender): Combine all sauce ingredients in the cup of a personal blender except for the chili, then purée until very well mixed. Taste and adjust seasoning as needed, then transfer to a serving boil. Stir in 1 thinly sliced fresno chile pepper for heat, then serve alongside roasted lamb.
To make the herb sauce (in a food processor): Place thinly sliced shallot, ½ cup mint, ½ cup parsley, peeled garlic clove, and ½ tsp kosher salt in the bowl of a food processor fitted with blade attachment. Pulse until finely chopped. Add 3 Tbsp red wine vinegar, 1 tsp lemon zest, and 2 Tbsp fresh lemon juice, then pulse again. With the motor running, stream in ⅔ cup extra virgin olive oil until well mixed. Taste and adjust seasoning as needed. Stir in 1 thinly sliced fresno chile pepper for heat, then serve immediately alongside roasted lamb.