Italian Breakfast Strata - Fresh-Pressed Olive Oil

Ingredients

• 6 tablespoons extra virgin olive oil, divided use
• 8 ounces cremini or mixed mushrooms, cleaned, trimmed, and coarsely chopped
• 1 small onion, peeled and diced
• 1/2 red bell pepper, trimmed, seeded, and diced
• 9 eggs
• 3 cups whole milk
• 1 teaspoon ground mustard
• 1 teaspoon Italian seasoning or dried basil
• Coarse salt (kosher or sea)
• Freshly ground black pepper
• 8 cups day-old Italian bread, cut or torn into bite-size pieces
• 12 pitted green or black olives, coarsely chopped
• 1 1/2 cups freshly grated Parmigiano-Reggiano, divided use
• 1 1/2 tablespoons fresh lemon juice, or more to taste
• 8 ounces baby arugula, spinach, or mixed greens
• Cherry tomatoes, for garnish (optional)