If you can't find baby artichokes, use four regular (globe) artichokes. Trim and steam them, then quarter them lengthwise.
Heat a medium heavy pot over medium-high heat. Add fennel seed; cook, stirring, 1 minute. Transfer to a plate.
Add 1 tablespoon oil to pot; swirl to coat. Season chicken with 1/2 teaspoon salt; coat with flour and shake off excess. Add half of chicken to pot. Cook, turning, until golden brown on all sides, about 10 minutes. Transfer to a plate. Repeat with remaining oil and chicken.
Reduce heat to medium and add leek, garlic, and pepper flakes. Add vinegar and 1/2 cup broth, scraping up browned bits with a wooden spoon. Cook, stirring, until vegetables are tender, about 6 minutes.
Add remaining 1 1/2 cups broth, 3 cups water, fennel seed, chicken with any accumulated juices, and artichokes. Bring to a boil; reduce to a simmer. Cook until chicken is tender, about 10 minutes. Season with remaining 1/2 teaspoon salt. Divide arugula among 4 shallow bowls, ladle in soup, and serve.
