Peel and slice the carrots into 1/4-inch rounds, dice the onion, slice the celery about 1/4 inch thick, and mince the garlic. Strip the kale leaves from the thick center ribs, discard the ribs, chop the leaves into bite-size pieces, then rinse and drain well.
Heat the olive oil in a Dutch oven over medium heat. Add the onion, carrots, and celery and stir every minute or so for 6 to 8 minutes, until the onion is soft and translucent but not browned.
Add the minced garlic and dried thyme and stir constantly for about 1 minute, until fragrant. Watch the heat closely so the garlic does not burn.
Pour in the chicken broth and scrape the bottom of the pot with a wooden spoon. Add the bay leaves, salt, and pepper, nestle the chicken breasts into the broth so they are fully submerged, and bring just to a boil over high heat.
Reduce the heat to medium-low, partially cover, and simmer for 18 to 22 minutes, until the thickest part of a breast reaches 165°F on an instant-read thermometer. Simmer 3 to 5 minutes more if needed and check again.
Lift the chicken onto a clean cutting board with tongs and rest for 5 minutes, then shred into bite-size pieces with two forks or meat shredding claws. Keep the pot simmering gently on low.
Stir the chopped kale into the simmering broth in two or three handfuls, letting each wilt before adding the next. Simmer uncovered for 8 to 10 minutes, until tender but still bright green.
Return the shredded chicken to the pot and simmer for 2 to 3 minutes to warm through. Remove from the heat, fish out the bay leaves, stir in the lemon juice, and season to taste. Ladle into warm bowls and top with parsley and grated Parmesan.