Heat 1 tablespoon oil in a large nonstick skillet over medium-high heat. Add chicken and cook, turning once, until deep golden brown, about 5 minutes total. Transfer to a 5- to 6-quart slow cooker. Finely chop 6 garlic cloves and add to the slow cooker along with broth, leek, carrot, wild rice, salt and pepper.

Cover and cook until the chicken is cooked through and the rice is tender, 3 hours on High or 6 hours on Low. Transfer the chicken to a clean cutting board and shred into bite-size pieces; discard bones.

Add asparagus and peas to the slow cooker; cook on High, uncovered, until the vegetables are bright green and tender, 15 to 20 minutes. Stir in the shredded chicken, parsley, chives and lemon juice.

Halve the remaining garlic clove and rub the cut sides over both sides of toasts. Drizzle the toasts with the remaining 1 tablespoon oil. Serve the soup with the toasts.