In a large stockpot, warm the olive oil over medium heat. Add the onion and cook for 8 minutes until fragrant and beginning to soften. Reduce the heat to low and add the carrots, leeks, garlic, and thyme. Continue cooking, stirring occasionally, until the vegetables are tender and golden brown, 25 to 30 minutes.
Pour the broth over the sautéed vegetables and bring to a simmer. Add the pasta and cook for 6 to 8 minutes more until the pasta is cooked through.
Add the kale and the shredded chicken and cook until the kale has wilted and the chicken is warmed through. Remove the thyme sprigs.
Serve the soup hot and enjoy with a squeeze of lime or sriracha sauce.