Heat the oil and butter in a nonstick pan over medium heat; saute the mushrooms and onions until soft. Add the marsala and thyme; cook for 2 - 3 minutes while stirring. Add in the cream and spinach. Simmer until the cream reduces and thickens. Remove from the heat; stir in parmesan cheese and chicken.
Overlap the phyllo sheets according to the amount of filling you have on a parchment paper lined baking sheet. Place the chicken mixture into the center of the phyllo. Fold in the ends and very gently roll it up. Position it seam side down.
Bake in the oven until the phyllo is golden brown and crisp, about 20 minutes. Let sit for 3 - 5 minutes before slicing.