If your walnuts aren't toasted, warm a dry skillet over medium heat and toast the walnut pieces, stirring frequently, until fragrant and lightly browned, 3–4 minutes. Transfer to a small plate to cool.
Trim the beets and cut each into thin wedges or 1/4-inch slices depending on your preference; place them in a medium bowl.
Whisk the apple cider vinegar, olive oil, honey, Dijon and minced shallot in a jar or small bowl until emulsified; season to taste with salt and pepper.
Pour about 2 tablespoons of the dressing over the sliced beets and toss gently. Let them sit for 5 minutes so they pick up a bit of flavor while you assemble the rest.
Roughly shred or slice the rotisserie chicken into bite-size pieces. If you prefer warm chicken, microwave briefly (15–30 seconds) or reheat in a skillet just until warmed through.
In a large bowl, drizzle 2–3 tablespoons of the remaining dressing over the mixed greens and toss gently to coat evenly. Taste and add more dressing if needed.
Divide the dressed greens among four plates or bowls. Top each with shredded chicken, dressed beet slices, diced avocado, pomegranate arils and toasted walnuts. Finish with a grinding of black pepper and an extra drizzle of dressing if desired.