Whisk the citrus juices, honey, vinegar, cumin and chili in a bowl. Slowly drizzle in the oil while whisking until emulsified. For a creamier dressing, blend everything in a high-powered blender. Taste for seasoning, adding more salt, honey and/or acid to suit your taste. Set aside.
Heat oven to 425 degrees.
Put 2 beets in a small oven proof skillet and add about 1/4-inch water. Cover the pan tightly with foil and roast until tender, 30 - 40 minutes. Cool, slip off the peel and slice into wedges.
Peel and slice the remaining beet into thin circles on a mandolin or with a sharp knife. Add to a mixing bowl.
Shave the carrots into ribbons with a vegetable peeler and add to the bowl with the beet rounds. Add the olive oil and a pinch of salt and toss to coat. Transfer the mixture to a large rimmed baking sheet and spread out in an even layer. Roast until golden brown, 12 minutes or so, stirring around halfway through.
Assemble the salad
Combine the roasted beet wedges, beet-carrot mixture, radicchio, red onion, scallion and chickpeas in a large bowl. Add 1/4 cup of the dressing and toss to coat.
Arrange the lettuce on a serving platter or a bowl and top with the dressed vegetables. Sprinkle the cheese, pistachios and herbs over the salad and gently toss. Drizzle the salad with little more dressing and serve.
You can roast the beets and carrots one or 2 days ahead.
Keep extra dressing in the fridge for up to a week. It may separate - just whisk before using.
Depending on how juicy your fruit is, count on 1 or 2 lemons/limes plus 1 orange to make the dressing.