Place the oven rack in the lower third of the oven; preheat to 425°.
In a food processor, pulse 2 cups mint leaves, the garlic, EVOO, half the lemon juice, 1 tbsp. salt, and 1 tsp. pepper until finely chopped and pastelike, about 6 pulses, stopping to scrape down sides as needed.
If needed, remove the lamb shoulder from netting without cutting through it; reserve netting. Using your hands, smear the paste all over the meat. Tie with kitchen twine or place back in the netting. Transfer the lamb to a small roasting pan lined with aluminum foil. Roast until an instant-read thermometer registers 145°, 60 to 70 minutes. Let the lamb rest while you make the sauce.
Bring blackberries, the remaining lemon juice, the sugar, and chopped mint to a simmer in a small saucepan over medium, mashing blackberries with a fork. Simmer, stirring often, until the liquid becomes syrupy, about 10 minutes. Remove the sauce from the heat and season with salt and pepper.
Remove twine or netting, slice the lamb, and arrange on a platter. Drizzle with the blackberry sauce.