In a medium pot, combine chicken and enough water to cover by 1 inch; season with salt. Bring to a simmer over medium-high, then reduce heat to a low simmer and cook chicken until cooked through, 18 to 20 minutes. Transfer chicken to a plate and let sit until cool enough to handle. Remove skin and bones and thinly slice.
Meanwhile, whisk together orange zest and juice, and oil; season with salt and pepper. Arrange beets on a large platter. Top with greens, chicken, apples, and pomegranate seeds. Drizzle with orange dressing.