Stir the yeast, sugar, and water in the bowl of an electric mixer fitted with the dough hook attachment. Let stand until mixture is foamy, 4 to 5 minutes.
Add oil, flour, and 2 teaspoons salt to yeast mixture. Mix on medium-low until dough comes together, 3 to 5 minutes. Raise speed to medium, and knead until dough is very smooth, about 5 minutes.
Transfer dough to an oiled medium bowl. Loosely cover with plastic wrap. Let dough rise in a warm, draft-free place until doubled in bulk, about 20 minutes.
Preheat oven to 400 degrees. Generously oil a 12-by-17-inch rimmed baking sheet. Spread dough into sheet. Flip dough to coat both sides with oil, and stretch dough into corners of sheet. Lightly cover with oiled plastic wrap. Let rise in a warm, draft-free place until dough comes to just above rim of sheet (3/4 inch) and is very billowy, about 30 minutes.
Uncover dough; sprinkle lightly with salt. Sprinkle evenly with cheese, walnuts, and rosemary. Top with caramelized potatoes, overlapping and pushing them into dough in places. Bake 5 minutes. Rotate sheet; bake 15 minutes. Rotate sheet again; bake until edges of crust are golden, about 5 minutes. Drizzle with oil. Let cool slightly before serving.
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