1. Preheat a charcoal grill to medium, ash-covered coals or a gas grill to 300° to 350° (medium) heat. Toss bell pepper, mushrooms, and onion with 1 Tbsp. vinaigrette, 1/4 tsp. salt, and 1/4 tsp. pepper.
2. Grill steak and vegetables at the same time. Grill steak on a charcoal grill, without grill lid, 13 to 16 minutes, or on a gas grill, covered with grill lid, 8 to 13 minutes. Grill vegetables in a lightly greased grill basket, stirring occasionally, on a charcoal grill, without grill lid, 10 to 12 minutes or until tender, or on a gas grill, covered with grill lid, 8 to 10 minutes or until tender.
3. Remove steak and vegetables from grill. Sprinkle steak with remaining 1/2 tsp. salt and 1/2 tsp. pepper. Let steak stand 5 minutes. Carve steak into thin strips.
4. Toss spinach with remaining 1 Tbsp. vinaigrette. Divide spinach among 4 bread slices. Top with cheese, steak, vegetables, and remaining bread slices. (Sandwiches will be very full.)
5. Cook sandwiches, in batches, in preheated panini press 2 to 3 minutes or until light golden brown and grill marks appear.