Preheat oven to 400°. Scrub beets, pat dry, and put on a rimmed baking sheet. Toss with 1 tbsp. oil and 1/2 tsp. salt. Bake until tender when pierced, 50 minutes. Roast nuts on another sheet, stirring often, until deep golden, 6 minutes.
Let beets cool. Meanwhile, toast cumin in a small frying pan over medium heat, stirring, until a shade darker, 2 minutes. Coarsely grind in a mortar or coffee grinder. Pour into a large bowl and whisk with remaining 1/2 cup oil, 3/4 tsp. salt, the lemon zest and juice, shallot, and pepper.
Trim beets and rub with a paper towel to remove peels. Cut into wedges, putting dark ones in a medium bowl and lighter ones in another medium bowl.
Toss chicken with light beets and enough dressing to coat. Toss dark beets with some dressing. In large bowl, toss greens and nuts with remaining dressing, then gently combine with chicken mixture. Arrange on a platter and tuck in dark beets. Season with more salt to taste if you like.