Pound chicken breasts flat between two pieces of plastic wrap. Cut them in half then sprinkle with salt and pepper, using 1 teaspoon salt & pepper total for all the chicken.
Spread a rounded teaspoon of Dijon on one side of each breast, then add a slice of mozzarella on top and roll up (if the cheese is too large for the chicken breast then cut it to fit). Use toothpicks to ensure the chicken stays together while cooking.
Dredge the chicken in flour, then heat a saute pan and add olive oil. Turn down to medium-high, then add chicken and saute about two minutes per side (don't overcrowd the pan; if necessary cook the chicken in batches). Remove chicken from the pan and tent with foil.
In the same pan, sauté mushrooms for 4 minutes, then add vermouth. Bring to a simmer then add onions, tomatoes, and parsley. Season with 1/2 teaspoon salt and paper.
Add the chicken back to the sauté pan and simmer covered for about 10 minutes, or until chicken temp reads at least 165.
Serve over the pasta of your choice (I use angel hair), and garnish with chopped onions and chopped parsley.