Heat a gas or charcoal grill to medium-high (about 450°). Trim off the most fibrous base of the kale stalks. Grill papaya slices until grill marks appear and papaya is slightly caramelized, about 3 minutes per side. Set aside. In a medium bowl, combine kale, olive oil, and salt and toss to coat. Grill kale until slightly charred and partially wilted, about 3 minutes per side. Remove and set aside.
Place a medium cast-iron pan on the grill or side burner. When pan is hot, add shallot and peanuts and cook until toasted but not burnt, about 5 minutes.
In a medium bowl, combine tamarind mixture, honey, olive oil, mustard, lemon juice, ginger, and salt. Whisk until thoroughly combined.
Transfer papaya rings to a serving platter, top with wilted kale, and drizzle with vinaigrette. Garnish with peanuts, shallots, and nasturtiums, if using.