Adjust oven rack to middle position and heat oven to 200 degrees.
Remove tenderloin from each chicken breast. Trim chicken breasts of excess fat. Halve each horizontally and pound 1/4" thick.
Season both sides of each cutlet with salt and pepper. Heat 2 teaspoons oil in 12" skillet over medium high heat until smoking.
Place 4 cutlets in skillet and cook without moving them until browned, about 2 minutes. Flip cutlets and continue to cook until second sides are opaque, 15-20 seconds. Transfer to large ovensafe plate.
Add 2 teaspoons of oil to now empty skillet and repeat to cook remaining cutlets. Cover plate loosely with foil and transfer to oven to keep warm while making pan sauce.
Add 2 teaspoons oil to empty skillet used to cook chicken and return panto medium heat. Add leek and cook, stirring often, until softened and browned around edges, 2-3 minutes. Add flour and cook, stirring constantly, 30 seconds. Add broth and vermouth, increase heat to medium-high, and bring to simmer, scraping pan bottom to loosen browned bits. Simmer rapidly until reduced to 3/4 cup, 3-5 minutes. Stir in any accumulated chicken juices, return to simmer and cook 30 secnds. Off heat, whisk in mustard, tarragon, and butter, season with salt and pepper. Spoon over cutlets and serve immediately.