Preheat a grill to 400°. Whisk olive oil, thyme, rosemary, salt, pepper, lemon zest, and garlic in a large bowl. Put 1 tbsp. marinade in another large bowl and set aside. Add chicken and bread to first bowl and toss to coat.
Thread alternating pieces of chicken and bread onto 8 (8in.) metal skewers, weaving prosciutto between pieces like a ribbon. Grill skewers, turning occasionally, until chicken is cooked through and bread is lightly charred, 10 minutes.
Whisk reserved marinade with lemon juice. Add kale and radishes and toss to coat. Serve skewers with salad.