Warm your oil of choice in a large stockpot over medium heat. Toss in the chopped onion, carrots, and celery, stirring now and then until they start to soften, roughly 3 to 4 minutes. Add the minced garlic during the last minute so it doesn't scorch and turn bitter.
Pour in the broth and drop in the bay leaf, then cover the pot and let everything come to a gentle simmer. Give it about 5 to 7 minutes so the vegetables continue softening. Once they're nearly tender, turn the heat down to low.
Scoop out roughly 1 1/2 cups of the hot broth and set it aside for the next step. Stir the cooked chicken and chopped kale directly into the pot so they can warm through and wilt gently in the simmering liquid.
In a separate bowl, whisk the eggs until they turn frothy and pale, about a minute of steady whisking. While still whisking, slowly stream in the reserved hot broth. This tempering step is what keeps your eggs silky instead of turning into scrambled bits.
Once the eggs are tempered, whisk in the fresh lemon juice until fully combined. You should have a smooth, pale yellow mixture that smells incredibly fresh and citrusy at this point.
Slowly pour the lemon-egg mixture back into the soup pot, stirring constantly as you go. Let it cook gently for another 3 to 5 minutes, but never let the pot come back to a full boil, or the eggs can curdle.
Taste the soup and adjust with salt and pepper as needed. Ladle it into bowls, then top with thin lemon slices and a sprinkle of fresh herbs before serving it warm.