In a small bowl, whisk together shallot, vinegar, orange juice, and oil; season with salt and pepper. Place beets in a small bowl. Pour 1/3 of the dressing over beets; toss to coat.
Place walnuts in a shallow dish. With your hands, form goat cheese into 12 equal balls. Roll balls in walnuts, turning to coat completely, then flatten into disks.
Place arugula and romain lettuce in a large bowl. Drizzle greens with remaining dressing, and toss to combine. Divide among plates, and top each serving with some of the sliced beets and 3 goat-cheese disks.