Set a large pot of salted water on the stove top. Bring to a boil. Stir in the tortellini and cook just under the set package instruction timeframe, about 5-6 minutes. Drain and rinse in cold water. Then toss with 1 teaspoon of olive oil so they don't stick together.
Place the basil leaves, remaining olive oil, red wine vinegar, garlic, and 1/2 teaspoon salt to the blender. Cover and puree until smooth.
Pour the tortellini, broccoli florets, tomato halves, chopped peppers, and scallions into a large salad bowl. Drizzle the dressing over the top and toss to coat. Then gently mix in the feta cheese. Taste, then salt and pepper as needed.
This salad can be made and refrigerated up to one week in advance.