Whisk together vinaigrette ingredients in a large bowl and pour half into a small bowl. Add beef and onions to large bowl, stir to coat, and chill 3 hours (or let sit at room temperature 2 hours), stirring once.
Heat a gas or charcoal grill to medium (350°). Oil cooking grate and grill onions until softened and lightly charred, turning once, 8 to 10 minutes. Increase heat to high (500°) and grill beef in 3 batches, letting heat come back up to 500° before starting a new batch. Cook, turning once, about 3 minutes total per batch, or until browned and cooked through (cut to test).
Toss greens and radishes in a large serving bowl with half of reserved dressing. Toss beef and sprouts with remaining dressing. Divide greens among large plates and top with grilled beef, onions, and sunflower sprouts.
Note: Nutritional analysis is per serving.