Preheat oven to 375°F. Brush bread with olive oil (use only 2 Tbsp.); sprinkle with salt and 1/4 tsp. pepper. Bake until lightly browned, 12 minutes. Remove from oven and rub each crostini with cut side of garlic clove. Transfer to a platter; set aside.
Heat 1 Tbsp. olive oil in a skillet over medium-high heat. Add artichokes and sauté until golden, 5 minutes. Add roasted peppers and sauté 1 minute more. Remove from heat; place mixture in a bowl. Add cheese and arugula; toss to combine. Season with pepper and spoon mixture onto crostini. Drizzle with remaining 1 Tbsp. oil and serve. Garnish with shaved Parmesan, if desired.