Cold Roast Lamb with Marinated Eggplant

Pulse garlic, rosemary, cumin seeds, and 2 Tbsp. kosher salt in a food processor until finely chopped. Pour in 1/4 cup oil and pulse again until a paste forms. Place lamb on a rimmed baking sheet and rub paste all over, pushing some inside layers of meat. Let sit at room temperature 1 hour.
Place racks in upper and lower thirds of oven; preheat to 300°F. Slice eggplants in half lengthwise, then slice each half lengthwise into thirds. Working in batches and using 1/2 cup oil total, place a few eggplant spears in a large bowl and drizzle with some oil. Toss, gently massaging each piece to coat; season with kosher salt and pepper. (Because eggplants are very porous when raw, working with a few pieces at a time makes it easier to control where the oil ends up.) Transfer to another rimmed baking sheet.
Toss shallots and 1/4 cup oil in the bowl you just used for the eggplants, then spoon over eggplants. Cover baking sheet with foil, sealing tightly.
Place baking sheet with eggplant mixture on bottom rack of oven and baking sheet with lamb on top rack. Roast until an instant-read thermometer inserted into the thickest part of lamb (avoiding gaps) registers 140°F for medium, about 90 minutes (start checking after 1 hour). Remove lamb from oven and tent with foil. Let cool about 20 minutes.
Meanwhile, remove eggplant mixture from oven, uncover, and toss with a spatula. Increase oven temperature to 500°F and immediately place eggplant mixture on top rack. Roast, tossing halfway through, until nicely browned and very tender, about 20 minutes. Let cool.
Remove foil from lamb and roast on top rack until brown and crisp, about 8 minutes. Let cool.
Once eggplant mixture is cool, toss in a large bowl with Master Vegetable Marinade, cover, and let sit at room temperature at least 1 hour.
Just before serving, toss parsley into eggplant mixture and transfer to a platter. Slice lamb 1/4" thick; arrange on another platter. Drizzle lamb generously with oil and sprinkle with sea salt. Serve with Lemony Yogurt Sauce and/or Grilled Serrano Salsa Verde alongside.
Lamb can be seasoned 12 hours ahead; cover and chill. Let sit at room temperature 45 minutes before roasting. Lamb can be roasted and eggplant mixture can be marinated 1 day ahead; cover and chill separately. About 1 hour before serving, remove lamb and eggplant mixture from the fridge and let sit at room temperature to warm up a bit.