Preheat grill to medium-high, direct heat (Gas grill: Turn on all burners. Charcoal: Spread coals into even layer.)

Puree scallions, ginger,2Tbsp. vinegar, Worcestershire, 2 Tbsp oil, and allspice in a blender. Combine chicken and marinade in a bowl, season with salt. Let sit 10 minutes.

Meanwhile, make dressing: Whisk remaining 2 Tbsp vinegar, 2 Tbsp oil, and cumin in a small bowl. Season with salt. Toss spinach, cucumber, and radishes in a serving bowl.

Grill chicken until cooked through, flipping once, about 10 minutes. Toss salad with dressing.