Set up grill for direct and indirect cooking and heat to medium. Once hot, clean and lightly oil grates with canola oil.
Whisk together cream cheese, sour cream, ¼ cup Parmesan, lemon zest and juice, and garlic in a bowl. Season with salt and pepper. Fold in artichokes and spinach. Transfer to a 9-inch cast-iron skillet. Top with remaining ¼ cup Parmesan. Cover with aluminum foil.
Place skillet over indirect heat and cook, covered, 10 minutes. Remove foil and rotate skillet, keeping over indirect heat. Cook, uncovered, until bubbly and golden brown, 18 to 20 minutes. Remove from grill.
Drizzle bread with olive oil. Grill over direct heat, turning occasionally, until crisp, 20 to 30 seconds. Serve with dip alongside.
