In a large bowl, dissolve salt in 1 cup hot water. Stir in 3 cups cold water and add chicken. Cover and refrigerate for 30 minutes.
Meanwhile, make dressing: In a blender, combine strawberries, vinegar, basil, honey, and mustard and blend until smooth. Remove center piece from lid, turn blender speed to low and drizzle in ¼ cup olive oil until emulsified. Taste and season with salt and pepper.
Preheat grill or grill pan to medium. Remove chicken from brine and thoroughly pat dry, discarding brine. Brush chicken and onion with 2 tablespoons olive oil. Season chicken with pepper and onion with salt and pepper. If you’re using an outdoor grill, consider threading the onion pieces onto skewers to avoid them falling between the grates. Grill, turning with tongs, until onion is tender and charred in spots, about 7 to 10 minutes, and chicken is cooked through to 165 degrees F, about 4 to 6 minutes per side. Transfer both to a cutting board. Tent chicken with foil and let rest; chop onion and place in a serving bowl.
Add, lettuce, radishes, strawberries, and feta to bowl with onion and toss. Add avocado and 3 tablespoons dressing and gently toss, adding more dressing if desired. Slice chicken and place on top of salad. Drizzle more dressing over chicken, sprinkle seeds on top, and serve. Refrigerate any leftover dressing in an airtight container for up to a few days.