Leave root and 1-inch stem on beets; scrub with a brush. Place beets in a medium saucepan; cover with water. Bring to a boil. Cover, reduce heat, and simmer 30 minutes or until tender. Drain and rinse with cold water; drain. Cool. Trim off beet roots and stems; rub off skins. Cut beets into 1/4-inch-thick slices. Place beet slices in a large bowl; add peaches and onion.
Combine chopped basil, chopped shallots, Champagne vinegar, olive oil, sugar, salt, and pepper, stirring with a whisk. Drizzle over salad; toss gently to coat. Sprinkle with crumbled goat cheese.