Step 1

Preheat oven to 325°. Season lamb generously with salt and pepper. Heat oil in a large heavy ovenproof pot over high heat. Sear lamb on all sides, adjusting heat as necessary to prevent scorching, until deep brown, about 18 minutes. Transfer lamb to a platter. Add onions, celery, and carrots to pot. Reduce heat to medium-low and cook, stirring occasionally, until onions begin to caramelize, about 25 minutes.

Step 2

Add 4 cups water to pot. Bring to a simmer, scraping up brown bits with a wooden spoon; return lamb and its juices to pot. Add water just to cover lamb (10–12 cups, depending on size of pot). Add thyme, rosemary, and garlic; bring to a simmer, cover, and transfer to oven.

Step 3

Braise until lamb easily pulls apart when pierced with a fork, 4–4 1/2 hours. Let lamb cool completely in stock. Transfer lamb to a platter. Strain stock through a fine-mesh sieve set over a large bowl; discard solids. Return stock and lamb to pot. Let cool, cover, and chill overnight. DO AHEAD: Lamb can be made 2 days ahead. Keep chilled.

Step 4

Remove lamb from stock; shred into large pieces. Strain stock through a fine-mesh sieve into a large saucepan; discard solids. (Because the fat is cold and still solid, it will strain easily.) Bring strained stock to a boil over medium-high heat. Reduce heat to medium; simmer until reduced to 3 cups, about 25 minutes. DO AHEAD: Lamb can be shredded into large pieces and stock can be strained and reduced up to 1 day ahead. Cover separately; chill.

Step 5

Preheat oven to 300°. Line a rimmed baking sheet with foil. Using a fork, poke holes into flesh all over squash; place on baking sheet. Coat squash lightly with oil and season with salt and pepper. Roast squash until juices seep from the holes and a paring knife inserted into the skin slides through easily, about 1 hour. Let squash cool; divide into quarters. Discard stem, seeds, and membranes. Using a spoon, scoop out 1" pieces of flesh; discard skin. DO AHEAD: Squash can be roasted 2 days ahead; cover and chill.

Step 6

Heat butter in a small heavy saucepan over medium heat, stirring often, until deep nutty brown (do not burn), about 8 minutes. Strain brown butter through a coffee filter–lined sieve set over a large skillet. Heat butter over medium heat; add garlic and cook until light golden, about 30 seconds. Using a slotted spoon, transfer garlic to a plate. Cook shallot in same pan until golden and beginning to get crisp, 2–3 minutes. Transfer shallot to plate with garlic.

Step 7

Reduce heat to medium-low and add onions. Cook, stirring occasionally, until well caramelized, about 45 minutes. Stir in reserved garlic and shallot, bay leaf, orange zest, cardamom, cinnamon, cloves, and turmeric. Season to taste with salt. DO AHEAD: Onion sauce can be made 3 days ahead. Cover; chill. Rewarm before using.

Step 8

Preheat oven to 350°. Place squash on a rimmed baking sheet and drizzle with 1/2 cup onion sauce. Bake until hot, about 10 minutes.

Step 9

Bring lamb, reduced stock, and 1/4 cup onion sauce to a simmer in a large pan over low heat. Divide lamb and squash among plates. Drizzle with reduced stock and remaining onion sauce; garnish with parsley.