Thoroughly trim steak of connective tissue and excess fat; put steak in a shallow, flat, nonreactive dish large enough to hold it flat. In a mixing bowl, stir together soy sauce, mirin, ½ cup chopped scallions, ginger, red pepper, and garlic; pour over steak, cover with plastic wrap, and place in the refrigerator to marinate for 1 to 2 hours.
Prepare a fire in an outdoor grill or preheat a broiler, stovetop ridged grill pan, countertop grill, or broiler.
Remove steak from marinade. Discard marinade. Pat steak dry with paper towels and season on both sides with black pepper. Grill or broil until medium rare, about 5 minutes per side. Remove steak from grill and let rest in a warm place, covered with foil, for about 10 minutes.
Meanwhile, in a large bowl, combine spinach and radicchio and toss well. Arrange beds of salad mixture on 4 serving plates.
Carve steak diagonally across the grain into thin slices. Arrange steak slices on top of beds of salad and garnish with dates, carrot, radishes, and diagonally sliced scallions. Drizzle salad or dot the plate surrounding it with some cilantro-mint-yogurt dressing. Pass remaining dressing at the table.
For the cilantro-mint-yogurt dressing: In a blender or food processor, combine vinegar, mint, cilantro, parsley, honey, and ginger. Blend or process until herbs are finely chopped and a smooth paste has formed. Put yogurt in a bowl and stir herb-ginger mixture into yogurt. Season to taste with salt and pepper. Use immediately or store in refrigerator, covered, and use within 3 to 4 days.