Bring a large pot of salted water to a boil. Add fettuccine and cook until al dente, just tender. Drain and set aside 1/2 cup (120 milliliters) of the pasta water.
While the fettuccine cooks, make the sauce. In a large saucepan, drizzle the olive oil over medium heat.
Add shallots and cook, stirring often, until softened.
Stir in the garlic and cook just until fragrant, about 30 seconds.
Add the mushrooms, then the oregano leaves.
Cook, stirring occasionally, until mushrooms darken. Pour in the heavy cream. Season to taste with salt and pepper.
Once heated through, use an immersion or stand blender to puree until smooth. If too thick to blend, 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 and the remaining ingredients are ready.
Toss the fettuccine with the mushroom cream sauce until coated. If needed, slowly add the reserved pasta water until desired texture.
Gently fold in the roasted vegetables, including any liquid.
Slice the cooked chicken breasts into strips.
Serve the fettuccine immediately with chicken and garnish with grated Parmesan cheese.