Preheat broiler for high heat. Line a baking sheet with foil.
Mix the goat cheese with the basil in a bowl. Spread cheese mixture evenly on four slices of bread. Place on prepared baking sheet.
Cook under preheated broiler until cheese has begun to turn golden brown, 3 to 5 minutes. Remove and set aside.
To make the vinaigrette, whisk together the mustard and vinegar in a small bowl until blended; season with salt and pepper to taste. Slowly pour in olive oil while continually whisking until dressing is smooth.
Place diced beets and mixed greens into a large bowl. Drizzle vinaigrette over top; toss to coat. Divide salad onto four chilled plates. Top with avocado and hazelnuts. Serve with goat-cheese tartines.
