Stem and coarsely chop tarragon. Peel, trim, and cut carrot into 1/2" diagonal slices. Rinse chicken breasts, pat dry, and season both sides with 1/4 tsp. salt and 1/4 tsp. pepper.

Heat 1 Tbsp. olive oil in a large non-stick pan over medium-high heat. Add carrots and pearl onions to hot pan and cook until vegetables begin to brown, 3-4 minutes. Whisk together miso, sugar, and 1/3 cup water in a small bowl.

Add miso-sugar mixture to pan with vegetables, cover, and reduce heat to medium. Cook until carrots are tender, 4-6 minutes. Remove cover, raise heat to high, and cook, stirring frequently, until liquid reduces to a glaze that clings to vegetables, 2-3 minutes. Sprinkle with half the tarragon (reserve remaining for sauce). Cover and set aside. While carrots cook, start chicken.

Heat 2 tsp. olive oil in a medium non-stick pan over medium-high heat. Working in batches, add chicken breasts to hot pan and cook until browned on one side, 3-5 minutes. Flip, cook 1 minute, then remove to prepared baking sheet. Repeat with remaining chicken breasts, replenishing olive oil as needed. Transfer baking sheet to oven and roast until chicken reaches a minimum internal temperature of 165 degrees, 7-8 minutes. Reserve pan; no need to wipe clean.

Return pan used to sear chicken to medium-high heat. Add evaporated milk and 2 tsp. tarragon to pan. Bring to a boil, reduce heat to medium, and simmer until thicken enough to coat the back of a spoon, 2-4 minutes. Season with 1/4 tsp. salt and a pinch of pepper. Taste, and add more tarragon if desired. Sauce will thicken as it cools.

Spoon sauce on plate and serve chicken over sauce. Arrange carrots and pearl onions next to chicken.