Dressing: Whisk together oil, vinegar, garlic, sugar, oregano, basil, salt and red pepper flakes and anchovies (if using). Cover and refrigerate if leaving more than 2 hours.
Salad: Thinly slice onion and peppers and place in large salad bowl. Cut tomatoes into wedges and squeeze out seeds and juice; coarsely chop and add to bowl. Whisk dressing and pour about half over salad; toss. (If making ahead, this can be left at room temperature for up to 2 hours for tomatoes to marinate.)
Just before serving, tear lettuce; add to salad along with mushrooms and pepperoni. Whisk remaining dressing; pour over salad and toss. Stir in half the cheese. Sprinkle with remaining cheese and basil.
Garlic Pita Wedges: Brush pitas with olive oil, then sprinkle with garlic powder or brush with garlic butter. Slice into wedges. Place on baking sheet in 350°F (180°F) oven for 8 to 10 minutes.