Melt butter in a 12" skillet over medium. Cook bread crumbs until crisp, 4–5 minutes; transfer to paper towels to drain. Purée the spinach, roughly chopped chives, ice cubes, salt, pepper, and 2 cups water in a blender until smooth; strain sauce through a fine-mesh sieve into bowl and stir in yogurt. In a separate bowl, beat cream until stiff peaks form; fold in yogurt mixture. Chill soup until ready to serve.
Heat a charcoal grill or set a gas grill to medium-high. (Alternatively, heat oven broiler.) Toss scallions with oil, salt, and pepper on a baking sheet; grill, turning as needed, until slightly charred, 1–2 minutes. To serve, divide soup between bowls. Garnish with reserved bread crumbs and minced chives; top with grilled scallions and grate horseradish over the top.
