Heat a grill to medium-high (about 450°). Brush chicken with 1 tbsp. oil and season with 1/2 tsp. each salt and pepper. Grill chicken, turning once, until just cooked through, 4 to 5 minutes total. Transfer to a plate. Grill kale until edges are charred and kale is softened, about 2 minutes.
Whisk remaining 1/4 cup oil in a large bowl with tahini, lemon zest and juice, honey, garlic, and remaining 1/2 tsp. each salt and pepper.
Roughly chop chicken, thinly slice kale crosswise, and add both to bowl of dressing. Add apples, almonds, and dates and toss gently but thoroughly.