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1
Preheat oven to 425u00b0F. Toss tomatoes, grated garlic, and 1 tablespoon 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.
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2
Reduce oven temperature to 350u00b0F. Toast hazelnuts on another baking sheet, tossing once, until golden brown, 10-12 minutes. Let cool; coarsely chop.
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3
Reduce oven temperature to 275u00b0F. Season pork with salt and pepper. Heat remaining 2 tablespoons oil in a large ovenproof pot over medium-high. Cook pork in batches, turning occasionally, until brown, 8-10 minutes; transfer to a platter.
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4
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, 2-2 1/2 hours.
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5
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.
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6
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.
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7
Sugo can be made 1 week ahead. Let cool; cover and chill, or divide in half and freeze up to 3 months.
