Preheat the oven to 500°. Preheat a pizza stone on the bottom of the oven for 45 minutes, or generously oil a large baking sheet. Squeeze the lemon halves into a large bowl of water and drop them in. Peel the sunchokes and add them to the water as you go (to prevent them from darkening). Cut the sunchokes into 1-inch pieces and return them to the lemon water.
Drain half of the sunchokes, transfer them to a medium saucepan and cover with the milk. Bring to a boil and simmer over moderate heat until tender, about 15 minutes. Using a slotted spoon, transfer the sunchokes to a blender. Add 1/2 cup of the milk and puree. Return the puree to the pan and cook over moderately high heat, stirring, until reduced to 1 cup, about 3 minutes. Season with salt and pepper.
In a large, deep skillet, heat 2 tablespoons of the oil. Add the onion, cover and cook over moderately low heat, stirring occasionally, until softened, 7 minutes. Add the cabbage, cover and cook, stirring occasionally, until tender, 12 minutes. Season with salt and pepper.
In another large skillet, heat the remaining 2 tablespoons of the oil. Drain the remaining sunchokes and pat dry. Add them to the skillet and cook over moderate heat, turning a few times, until crisp, 6 minutes. Season with salt and pepper and transfer to a plate.
Cut the pizza dough in half; work with 1 piece at a time and keep the rest covered. Lightly flour a pizza peel or a flat baking sheet. On a lightly floured work surface, roll out the dough to form a 12-inch round. Transfer the round to the peel.
Spread half of the sunchoke puree over the round, leaving a 1/2-inch border. Scatter half of the cabbage over the puree, followed by half of the crisped sunchokes. Sprinkle half of the Emmental on top. Slide the pizza onto the hot stone or oiled baking sheet and bake on the bottom of the oven for about 4 minutes, until crisp on the bottom and bubbling on top. Sprinkle the pizza with half of the pecorino and thyme, cut into wedges and serve. Repeat with the remaining ingredients to make a second pizza.
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