Preheat a grill to medium. Season the chicken with 1 teaspoon allspice, 1/4 to 1/2 teaspoon cayenne, and salt and black pepper to taste; transfer to a large bowl and toss with 1 garlic clove, 2 tablespoons olive oil and half of the lemon juice. Toss the plums with 1 teaspoon olive oil, the remaining 1/4 teaspoon allspice and a pinch each of cayenne and salt in another large bowl.
Grill the chicken, turning occasionally, until marked, 10 minutes. Cover and continue grilling until cooked through, 15 to 20 minutes. (Move the chicken to a cooler part of the grill if it's getting too dark.) Grill the plums, flipping, until slightly softened, 5 minutes. Transfer the chicken and plums to a platter; brush the chicken with the honey.
Meanwhile, bring a large pot of salted water to a boil. Add the broccoli and cook until crisp-tender, about 3 minutes. Drain and rinse under cold water; pat dry and transfer to a medium bowl. Add the remaining 1 tablespoon olive oil and the almonds; season with salt and black pepper and toss to coat.
Whisk the yogurt with the remaining grated garlic and lemon juice in a small bowl; season with salt. Serve the chicken with the plums and broccoli; spoon the yogurt sauce over the broccoli.