Heat oil in a large pot over medium-high heat. Set a little bit of the white part of the leek aside to top your soup later and add the remaining leek and all of the garlic to the pot. Season with salt and pepper. Cook, stirring occasionally until the leek and garlic are totally softened and starting to get a little color, 8–10 minutes.
Add tomatillos (or tomatoes) and season again with salt and pepper. Cook, stirring occasionally until tomatillos (or tomatoes) are totally falling apart, almost turning to a jam-like texture. Add leafy greens and season again with salt and pepper. Cook them just till wilted, a few minutes.
Add chicken broth and bring to a simmer. Taste every now and then for seasoning and simmer until the broth is tangy, a little salty, and very delicious (season with vinegar, lemon or lime juice if using tomatoes). If you can’t taste anything because you have a cold, this will take about 10–15 minutes of simmering.
Add the chicken and season one more time to taste. Divide among bowls and top with a few rings of the raw leek, all the herbs, a drizzle of olive oil, and some crushed chili flakes or hot sauce if you like.
