Bring a large pot of water to a boil; add salt and beet. Cook pasta 2 minutes less than instructed on package (beet should be knife-tender). Drain; when beet is cool enough to handle, chop into small pieces.
Place prosciutto in a large saute pan over medium-high; cook, stirring occasionally, until crisp, about 3 minutes. Transfer to a plate.
Swirl oil into pan and add onion. Cook, stirring occasionally, until softened, 3 to 4 minutes. Add cabbage and cook until slightly tender, about 2 minutes. Stir in beet. Increase heat to high, pour in wine, and cook, stirring frequently, until reduced by half, about 2 minutes.
Add pasta and toss to combine. Add prosciutto and season with salt. Serve immediately, with cheese.