How to Make It
Preheat the oven to 350°. Season the goat with salt and pepper. Heat a large, enameled cast-iron casserole. Add the fennel and coriander seeds and the peppercorns and toast over moderate heat until fragrant, about 1 minute; transfer the spices to a plate to cool. Add the oil to the hot casserole and swirl to coat the bottom. Add the goat and cook over moderately high heat until richly browned all over, about 4 minutes per side. Transfer the goat to a plate.
Add the onions, carrots, celery and fennel to the casserole and cook over moderate heat, stirring occasionally, until softened, about 8 minutes. Add the wine and boil over high heat until reduced by half, scraping up any browned bits on the bottom of the casserole, about 3 minutes. Add the chicken stock and bring to a boil. Return the goat to the casserole. Add the bay leaf, rosemary sprig and the 1/2 cup of whole olives. Put the toasted spices and the crushed red pepper in a tea ball and submerge it in the liquid. Cover and bake, turning the goat once, until the meat is very tender, about 3 hours. Let the goat cool to room temperature in the cooking liquid.
Discard the bay leaf, rosemary sprig and the contents of the tea ball. Transfer the goat shoulder to a large rimmed baking sheet and remove all of the meat, breaking it up into 2-inch pieces. Discard any fat, bones and gristle
Pass the vegetables and cooking liquid through the coarse disk of a food mill and return the sauce to the casserole. Add the goat meat and the remaining 1/2 cup of chopped olives. Simmer over low heat for 3 minutes. Stir in the mint leaves and lemon juice and season with salt and pepper.
Bring a large pot of salted water to a boil. Add the spaghetti and cook until almost al dente. Drain the spaghetti and add it to the ragù in the casserole. Bring to a simmer and cook for 2 minutes, stirring gently. Add the olive oil, toss well and serve.