Heat the olive oil and rosemary sprigs in a small saucepan over medium-low heat until just beginning to bubble. Remove from heat and cover for 10-15 minutes. Transfer to a heat-proof jar while you prepare the rest of the ingredients.
Preheat the oven to 400°F.
Thinly slice the potatoes using a mandoline set to one notch below ⅛-inch (or in a food processor or by hand). Toss the potatoes in 1 tablespoon of the rosemary olive oil and ½ teaspoon of salt. Spread the potatoes out in a thin layer on a large parchment-lined baking sheet. Place the garlic cloves on one end of the baking sheet and bake for 16-20 minutes until the potatoes just begin to soften and the edges turn brown. Remove from heat and set the potatoes aside.
While the potatoes cook, prepare the shallots. Thinly slice the shallots using a mandoline or by hand. Heat ½ teaspoon of the rosemary olive oil in a medium skillet over low heat and place a cup full of water nearby. Add the shallots to the skillet, breaking them apart with a spoon, with a pinch of salt. Cook, stirring, until the shallots begin to caramelize. Stop stirring and increase the heat to medium-low. Once a brown glaze has formed on the bottom of the skillet, pour 1 tablespoon of water into the pan to deglaze it, using a spatula to scrape up any brown bits and stirring them into the shallots. Repeat this process until the shallots have been thoroughly caramelized, 8-12 minutes. Transfer to a bowl and set aside.
Toss the shaved brussels sprouts with 1-2 teaspoons rosemary olive oil, apple cider vinegar, and pinch of salt and pepper. Set aside to marinate.
Once the potatoes and garlic are done, peel and mince the roasted garlic. Increase the oven temperature to 450°F.
Place your par-baked pizza crust on a pizza stone or pizza pan.
Brush 2 teaspoons rosemary oil over the crust and sprinkle the minced roasted garlic on the crust as evenly as possible.
Lay the torn mozzarella all over the crust as evenly as possible. Top with caramelized shallots then sprinkle with ¾ of the grated gruyère. Top with roasted potatoes, overlapping as needed. Sprinkle the remaining gruyère over the top, then grate the parmesan cheese over the potatoes. Bake for 5-8 minutes until the cheese melts and the crust begins to turn golden brow.
Remove from oven, transfer to a board for slicing. Slice and top each spice with brussels sprout slaw.