Eggplant, Red Pepper, and Fontina Panini with Spinach Salad

Brush panini press, panini pan, or grill rack with oil; heat press, pan, or barbecue (medium-high heat). Brush eggplant rounds and pepper strips with oil; sprinkle with salt and pepper. Working in batches, grill eggplant until tender, turning occasionally, about 6 minutes. Working in batches, grill bell peppers until tender, turning occasionally and flattening with back of spatula for even cooking, about 6 minutes. Place 8 eggplant rounds and 8 bell pepper strips on rimmed baking sheet. Place remaining eggplant and peppers on work surface and slice thinly; reserve for salad.

Place 4 bread slices on work surface. Divide half of Fontina cheese among bread slices; top each with 2 eggplant rounds, 2 tablespoons Parmesan cheese, 1 tablespoon basil, and 1 teaspoon oregano. Top each with 2 bell pepper strips. Arrange remaining Fontina cheese atop peppers, then top with remaining bread slices, pressing to compact. DO AHEAD: Can be made 2 hours ahead. Cover with plastic and let stand at room temperature.

Heat panini press, panini pan, or barbecue (medium-high heat). Brush press or pan with oil (if using barbecue, lightly brush bread slices with oil). Place sandwiches in press or pan or on grill; cover and cook until bread is golden and cheese is melted, about 5 minutes per side.

Meanwhile, whisk 2 tablespoons extra-virgin olive oil and vinegar in large bowl; season to taste with salt and pepper. Add spinach, thinly sliced eggplant, and thinly sliced peppers; toss to coat. Season salad to taste with salt and pepper.

Cut sandwiches in half. Serve with spinach salad.