Whisk together first 5 ingredients. Cover and chill until ready to use (up to 24 hours).
Toss together greens and basil. Top with pears, oranges, avocados, and blue cheese; toss. Drizzle with vinaigrette. Serve immediately.
It's ideal to slice pears and avocados at the last minute so they won't turn brown. You can do this one hour ahead and toss in a small amount of lemon juice. (Too much will alter the flavor of the vinaigrette.)