Line a rimmed baking sheet with parchment paper and lay prosciutto slices on top, spaced so they aren’t touching. Bake until slices start to shrivel and turn golden, about 10-12 minutes. Keep your eye on it, as the prosciutto can burn quickly. Let cool. Crumble into pieces.
Heat oil over medium-high heat. Cook mushrooms, tossing occasionally, until browned and tender, 5-8 minutes. Reduce heat to medium-low. Add shallots and 2 teaspoons chives. Season with salt and pepper, and cook, stirring often, until shallots are translucent and softened, about 2 minutes. Add garlic and cook for 1 minute. Add broth and reduce heat to low. Bring to a simmer and cook until only a thin layer of broth coats the bottom of the pot, 5-7 minutes.
Meanwhile, cook pasta in a large pot of boiling salted water, stirring occasionally, until just al dente, about 3 minutes less than package directions. If using fresh fettuccine, it will only take a few minutes to cook.
Using tongs, transfer pasta directly to pot with mushroom sauce mixture and add 1 cup pasta cooking liquid. Crumble half the prosciutto into the pot. Increase heat to medium, bring to simmer, and cook, tossing constantly, until liquid is slightly thickened, about 2 minutes. Add cream, return to simmer, and cook, tossing, until pasta is coated, about 1 minute. Remove from heat, add butter, and toss to combine.
Divide pasta among bowls. Top with remaining prosciutto and chives.