Cook the pasta according to the instructions on the package. Drain well and set aside.
Season the chicken with salt and pepper. Dip into the egg and then coat well with the breadcrumbs, pressing them on tightly.
In a non-stick pan over medium heat, melt the butter, then add onion, carrot, celery, and mushrooms and cook until soft (about 5 minutes). Pour in the wine and simmer until the wine has mostly evaporated.
Add broth and cream and season with thyme, salt, and pepper. Simmer until the sauce reduces and thickens (about 10 minutes). Stir in the lemon zest and remove the thyme sprigs.
In the meantime, heat the oil in a cast iron skillet or non-stick pan over medium heat. Cook the chicken until golden brown, about 3 to 4 minutes per side, to an internal temperature of 160°F using a thermometer. Remove and set aside on a paper towel-lined plate. Slice before serving.
Once thick, toss the pasta in the sauce, serve with chicken and bread. Garnish with parmesan cheese.