Heat grill to medium. Once it is hot, clean the grill grate with a wire brush. Just before grilling, oil the grill grate with canola oil.
Combine the pistachios, parsley, capers, lemon juice, 2 tablespoons of the olive oil, and ¼ teaspoon each salt and pepper in a medium bowl; set aside.
Season the chicken with ¼ teaspoon each salt and pepper. Grill, covered, turning occasionally, until an instant-read thermometer inserted in the thickest leg (avoiding the bone) registers 165° F, 15 to 22 minutes.
Meanwhile, toss the broccoli, lemon slices, the remaining 2 tablespoons of olive oil, and ¼ teaspoon each salt and pepper in a medium bowl. Grill, turning occasionally, until the broccoli is golden brown and crispy, 6 to 10 minutes.
Spread the yogurt on a plate and serve the chicken and broccoli over the top, drizzled with the pistachio-herb sauce.