½cup(1 stick) chilled unsalted butter, cut into pieces
4ouncesGrana Padano cheese, finely grated, plus shaved for serving
1poundfresh or dried pappardelle
Preheat oven to 425°. Toss tomatoes, grated garlic, and 1 Tbsp. oil on a rimmed baking sheet. Arrange tomatoes, skin side down, in a single layer; season with salt and pepper. Roast until browned around the edges, about 25 minutes. Set aside.
Reduce oven temperature to 350°. Toast hazelnuts on another baking sheet, tossing once, until golden brown, 10–12 minutes. Let cool; coarsely chop.
Reduce oven temperature to 275°. Season pork with salt and pepper. Heat remaining 2 Tbsp. oil in a large ovenproof pot over medium-high. Cook pork in batches, turning occasionally, until brown, 8–10 minutes; transfer to a platter.
Add shallots and whole garlic cloves to same pot and cook, stirring occasionally, until browned, about 5 minutes. Add wine, scraping up any browned bits; bring to a boil and cook until wine is reduced slightly, about 5 minutes. Return pork to pot and add milk, broth, and reserved tomatoes. Bring to a boil. Tie oregano and thyme sprigs together with kitchen twine; add to pot. Cover and transfer to oven. Braise meat until very tender, 2–2½ hours.
Remove from oven; discard herbs and use a potato masher or pair of forks to shred pork and mash vegetables into medium-size pieces. Gradually add butter and grated cheese, stirring as you go to fully incorporate into sugo.
Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until al dente. Drain pasta, transfer to a platter, and toss with half of sugo. Serve topped with hazelnuts, chopped oregano and thyme, and more Grana Padano.