Place vegetables, garlic, salt, and pepper on a large
baking sheet. Mix well. Drizzle 1 tablespoon olive oil over the top of the vegetables and mix again. Roast for 20–25 minutes until tender. Allow to cool.
Grease a 9" × 13" casserole dish and lay 6 sheets of phyllo pastry inside. Sparingly brush with the remaining olive oil. Spread the Greek yogurt on top of the phyllo dough evenly. Sprinkle the shredded mozzarella, then the Cheddar, over the yogurt.
Pour the cooled vegetables on top of the cheese and spread evenly.
Crumble the remaining 2 phyllo sheets on top of the vegetable mixture. Sprinkle the Parmesan cheese and basil on top.
Bake for 15 minutes until the edges are golden brown. Allow to cool, and slice into 12 squares. Serve warm or cold.