Bring a large pot of salted water to a boil. Add fettuccine and cook until al dente, just tender. Drain.
While the fettuccine cooks, make the sauce. In a medium saucepan, drizzle olive oil over medium heat.
Add shallots and cook, stirring often, until softened. Add garlic and cook just until fragrant, about 30 seconds. Stir in the mushrooms, then the rosemary.
Cook, stirring occasionally, until mushrooms darken. Pour in the heavy cream. Use an immersion or stand blender to puree until smooth. If too thick, add a little more cream. If a stand mixer was used, return the sauce to the saucepan and place over low heat until heated through.
In the emptied large pot, toss the fettuccine with the roasted vegetables, including any liquid. Slice the chicken breasts into strips. Serve the fettuccine immediately with chicken topped with mushroom cream sauce and parmesan cheese.