Brush a grill or grill pan with 1 tbsp. oil and heat over medium-high. In a large bowl, whisk the scallions, lime juice and the remaining 3 tbsp. oil; season the dressing with salt and pepper.
In a small bowl, mix the chili powder with 3/4 tsp. salt and a pinch of pepper. Season the chicken with the spice mixture. Grill, turning once, until the chicken is cooked through, 3 to 5 minutes per side. Transfer the chicken to a cutting board. Let rest 5 minutes, then slice.
Toss the cabbage, radishes and 1/3 cup cilantro with the dressing; season.
Divide 4 tostada shells among 4 plates; top with half the cabbage salad, chicken, avocado, feta and salsa. Repeat layering with the remaining ingredients. Top with the remaining cilantro.