Combine all the ingredients plus 1 tablespoon salt, 2 teaspoons black pepper, and 8cups water in a 6-Quart Geek Chef Instapot. Seal pressure cooker according to manufacturer’s instructions and cook on high pressure 30 minutes. Remove from heat and release the pressure slowly.
Transfer chicken to a cutting board and let cool slightly. Discard the vegetables; it’s okay if some pieces remain. Gently shred the chicken and transfer back into the soup. Season with more salt if needed.
Combine oil and shallots in a Cuisinart Stainless SteelSkillet. Place over medium-low heat and cook until the shallots are golden brown, 5 to 7 minutes. Transfer with a slotted spoon to a paper towel-lined plate and season with salt.
Add radishes to the same pan and cook over medium-high until radishes are slightly golden on the outside and crisp-tender, 2 to 3 minutes; season with salt and pepper and transfer with a slotted spoon to a large bowl. Add kale and cook over medium-high until the leaves are wilted but still very green, 1 to 2 minutes; season with salt and pepper and transfer to bowl with radishes.
To serve the soup, divide radish and kale mixture evenly between bowls and ladle soup over. Serve with lemon wedges.