Trim excess fat from chicken, then season on both sides with salt, pepper and half the tarragon. Heat oil in a large, lidded sauté pan over medium-high flame—enough oil to cover the bottom of the pan, 2 tablespoons or so. Cook chicken skin side down until nicely browned, 5 to 7 minutes. Turn chicken and reduce heat to medium. Cook for about 5 minutes. Transfer chicken to a plate.
Add mushrooms to pan and cook, stirring occasionally, for about 5 minutes. Add leeks to pan, drizzling in a little more oil, if needed (mushrooms can be sponges). Toss to coat with oil and combine with mushrooms. Season with salt and cook, stirring occasionally, until leeks are slightly softened, about 5 minutes. Don’t let them brown—reduce heat, if needed.
Clear a space in the middle of the pan, drizzle in a little more oil, if needed, and add garlic and remaining tarragon to pan. Cook, stirring constantly, until fragrant, about 45 seconds.
Add wine and chicken broth to pan, scraping up any browned bits. Return chicken to pan, skin side up, nestling pieces into leek mixture. Bring liquid to a boil, then reduce to simmer. Partially cover pan and cook until chicken is cooked through, 25 to 35 minutes.
Transfer chicken to plate. Spoon a little of the pan liquid into the half & half to slightly warm it, adding a tablespoon at a time until you’ve added 4 altogether. This will keep it from curdling when you add it to the pan. Slowly pour the warmed half & half into the pan, stirring to combine. Return chicken to pan and cook, uncovered, for about 5 minutes to combine flavors. Taste and adjust seasoning with salt, as needed.
Meanwhile, cook the rice, trying to time it to be ready when the chicken is done.
If the pan liquid hasn’t sufficiently thickened into a sauce, mix the tablespoon of cornstarch in a small bowl with a tablespoon of cold water, stirring until it is completely dissolved. Warm the cornstarch mixture by adding a little of the pan juice at a time to the bowl. This will keep it from clotting. Once again transfer the chicken to the plate, tenting it with foil, and slowly stir the cornstarch mixture into the pan. Cook, stirring occasionally for 3 or so minutes, allowing the sauce to thicken.
Serve. Place some rice and a piece of chicken on each plate. Spoon some of the leek/mushroom mixture over the rice—and the chicken, if desired.