Preheat the oven to 375°F.
Place the cabbage, shiitake mushrooms, onions and miso in a large bowl. Add a tablespoon of the olive oil and, using clean hands, massage the miso into the veg, taking care to hit all the nooks and crannies.
Transfer the veg to a baking sheet, greased with another tablespoon of the olive oil. Create a space for the head of garlic and pop the sheet into the oven. Roast for 30 minutes, rotating the pan halfway through. Set aside until ready to use.
While the veg is roasting, place a large pot of water over high heat. Bring the water to a boil, salt it liberally and add the spaghetti. Cook according to the package's direction.
While the spaghetti is cooking, pour the remaining olive oil into a large, deep skillet. Add the anchovies and place over medium-low heat. Cook until the anchovy starts to disintegrate, about a minute or so. Stir in the shallots along with a pinch of salt. Saute until just translucent.
Add the Worcestershire and the mustard and toss to coat. At this point, your pasta should be al dente. Transfer the pasta from its cooking liquid to the skillet. Add in the roasted veggies and the butter and toss to coat, stirring in some of the pasta cooking liquid as needed. Stir in the lemon juice and taste. Season with salt accordingly.
Divide the pasta among 4 bowls and top with shredded Parmigiano Reggiano cheese. Serve immediately.