Pull some bread from ciabatta rolls to form slightly hollow centers. Drizzle ciabatta rolls with olive oil. Spread roll bottoms with whole grain mustard. Sprinkle roll bottoms with half of fontina cheese.
Heat 2 tablespoons oil in large skillet over medium-high heat. Add mushrooms; saute 4 minutes. Add chopped shallots and pressed garlic; saute 3 minutes. Add chicken; saute 2 minutes to heat through. Transfer to plate. Add 1 tablespoon oil to skillet. Add spinach; saute 2 minutes. Season to taste with salt and pepper. Drain.
Spoon chicken mixture, then spinach over roll bottoms. Top with remaining cheese. Cover with roll tops.
Wrap each sandwich tightly in foil. Bake sandwiches until cheese melts, about 20 minutes.