Place the beets in a medium saucepan and cover with water. Bring to a boil over high heat, then lower to a simmer and cook until the beets are tender, 30 to 40 minutes. Drain, let cool, peel, and dice into ½-inch/1.25 cm cubes.
In a large bowl, combine the arugula, lettuce, strawberries, beets, onion, most of the roasted nuts, and chili if using.
In a small jar or bowl, whisk together the reduced balsamic, olive oil, apple cider vinegar, lemon juice, honey, black pepper, and salt until emulsified.
Pour the dressing over the salad and toss gently to combine. Transfer to a serving bowl, scatter the remaining nuts and cheese on top, and serve immediately.