Set a large pot over medium heat and pour in olive oil. Wait until it shimmers a little. ( Note ) Don’t rush this part - you want it warm, not smoking.
Add chopped onion and garlic. Stir until soft and you can smell them. ( Note ) About 3–4 minutes. They should look golden, not brown.
Toss in celery and carrots. Stir a few times to coat them in the oil. ( Note ) Let them cook until they start to soften slightly.
Add the chicken pieces. Stir until they lose that pink color on the outside. ( Note ) They’ll finish cooking in the broth.
Pour in the chicken broth. Scrape the bottom of the pot with your spoon to lift any browned bits. ( Note ) That’s where the flavor hides.
Turn the heat down to low. Let it simmer gently for 20–25 minutes. ( Note ) You’ll see small bubbles around the edges - that’s perfect.
Add the chopped kale. Stir once or twice. ( Note ) It’ll wilt down in a few minutes and turn dark green.
Turn off the heat. Add lemon juice, salt, and black pepper. ( Note ) Taste it, then adjust seasoning as needed.
Serve warm. ( Note ) The soup thickens a bit as it cools - that’s normal.