Place the oil in a large pot over medium heat. When the oil is hot, add the leeks, carrot, and celery, season with salt and pepper, and cook, stirring occasionally, until the vegetables have become tender and lightly brown, about 4 minutes.
Add the broth and bring to a simmer. Add the chicken and rice, return to a simmer, and cook for 10 minutes, stirring occasionally.
Add the peas and asparagus, and simmer for 5 to 7 more minutes until the chicken and rice are cooked through and the vegetables are tender.
Notes
You can add any small pieces of vegetables you like to this soup. In this version, I used asparagus and peas. Little pieces of broccoli, sugar snap peas, and sliced string beans would also be good choices. If you want to add a denser vegetable, like cauliflower, for instance, add it earlier in the soup simmering process, as that will take longer to cook.
If you add precooked rice to the soup, reduce the amount of broth to about 6 cups.
Other herbs can be used in place of the thyme. Try tarragon, basil, oregano, or marjoram, and know that if you are using a dried herb, you want to add less of the herb than you would fresh, as the flavor is more concentrated.