Stir lemon juice into a medium bowl filled with water. Trim away tough outer leaves of the artichokes to expose their tender, pale green interior. (For illustrated, step-by-step instructions, see Trimming Baby Artichokes.) Put trimmed artichokes into lemon water; set aside.
Heat the oil in a dutch oven over medium-high heat. Add the garlic, mushrooms, and carrots and cook, stirring frequently, until soft, about 8 minutes. Season the chicken with salt and pepper. Increase heat to high and add the chicken and tomato paste; cook, stirring occasionally, until chicken is lightly browned, about 6 minutes. Add the vinegar and cook, stirring constantly, until liquid has evaporated, about 1 minute. Drain the artichokes and add them, along with 2 tbsp. tarragon and the chicken broth, to the pot. Bring broth to a boil and reduce heat to medium-low. Simmer until chicken and artichokes are very tender and the broth has reduced by half, about 40 minutes. Season with salt and pepper and set sauce aside.
Meanwhile, bring a pot of salted water to a boil over high heat. Add pasta and cook until al dente, about 7 minutes. Reserve 1 cup pasta water. Drain pasta and transfer to pot with artichoke and chicken sauce; set over high heat. Bring sauce to a boil, stir in 1⁄2 cup parmesan and cook, tossing occasionally with tongs, until sauce thickens and clings to pasta, about 2 minutes. (If sauce is too dry, pour in a little pasta water.) Add remaining tarragon and season with salt and pepper. Transfer pasta to 4 bowls and sprinkle with remaining parmesan.