Season chicken with 3/4 teaspoon salt and 1/2 teaspoon pepper.
Melt butter in a Dutch oven over medium heat. Working in batches, add chicken to the skillet in a single layer and cook until golden brown, about 7-8 minutes; set aside.
Add mushrooms and shallots, and cook, stirring occasionally, until tender, about 3-4 minutes.
Stir in garlic and thyme until fragrant, about 1 minute.
Whisk in flour until lightly browned, about 1 minute.
Stir in chicken stock and Dijon, scraping any browned bits from the bottom of the Dutch oven.
Stir in fusilli; season with salt and pepper, to taste.
Bring to a boil; reduce heat and simmer, stirring occasionally, until pasta is just al dente, about 8-9 minutes.
Stir in chicken, spinach, Parmesan and heavy cream until spinach has wilted and pasta is tender, about 3 minutes.