Season the chicken with ½ teaspoon salt, ½ teaspoon pepper, and paprika.
Heat olive oil in a large pot over medium-high heat.
Once hot, add the chicken breasts and sear until golden, about 3 minutes per side.
Remove the seared chicken from the pot and set aside on a clean plate.
Reduce the heat to medium. Add the butter to the pot and melt.
Once melted, add the mushrooms and saute for 5 to 8 minutes, or until tender.
Remove ⅔ cup of the sauteed mushrooms and set aside.
Add the garlic, shallots, ½ teaspoon salt, ½ teaspoon pepper, ½ teaspoon rosemary, ½ teaspoon thyme, and ½ teaspoon oregano to the pot. Stir to combine and saute for 2 minutes, or until the shallots are tender.
Deglaze the pot with wine and bring to a simmer. Simmer until the wine has mostly evaporated, about 3 minutes.
Sprinkle the veggies with flour and stir to combine the roux.
Cook the roux for 1 minute more, stirring frequently.
Add the half and half and chicken broth to the pot, whisk to combine, and bring to a low simmer. Simmer, whisking occasionally, for 10 minutes or until the sauce has thickened slightly.
Remove the pot from the heat and sprinkle with 1 cup mozzarella, ½ cup fontina, and ½ cup Parmesan. Whisk until melted.
Add the cooked noodles to the pot and toss to combine.
Twirl and transfer the noodles to the prepared casserole dish.
Ladle the noodles with any remaining sauce left in the pot.
Sprinkle the casserole with the remaining cheeses and the reserved sauteed mushrooms.
Nestle the seared chicken into the noodles.
Transfer to the oven and bake for 15 minutes, or until the chicken is cooked through and the casserole is bubbly.
Remove from the oven and sprinkle with the remaining rosemary, thyme, and oregano before serving.