For Dressing: Place lemon juice and vinegar in small bowl. Gradually whisk in ½ cup oil. Stir in cheese. Season with salt and pepper. (Dressing can be made 1 day ahead. Cover and chill.)
Preheat oven to 375°F. Heat remaining ⅓ cup oil in medium ovenproof skillet over medium heat. Add bread pieces; toss to coat. Add herbs and garlic; toss to coat. Sauté until bread is crisp, about 4 minutes. Using slotted spoon, transfer croutons to plate in single layer. Cool.
Add butternut squash to same skillet, tossing to coat with any remaining herbs and oil. Cover skillet with foil and transfer to oven. Toss beets with olive oil and place on in different roasting pan and cover with foil. Add to oven. Roast until butternut squash beets are tender, about 45 minutes. Cool squash and beets. Peel, if desired; cut in half.
You may toss beets and butternut squash together, but colors bleed.
Toss arugula with ½ cup dressing in large wide bowl. Season to taste with salt and pepper. Top with butternut squash, croutons and beets and serve.