Prep ingredients: Preheat oven to 450°F with a rack in the upper third. Peel and finely chop half of the shallot. Add walnuts to a baking sheet; bake until lightly toasted, 3 minutes (watch closely). Transfer to a cutting board and coarsely chop. In a medium bowl, combine 1 teaspoon each of vinegar and water, 1 tablespoon oil, and 2 teaspoons of the shallots. Season with salt and pepper.

Mash potatoes: Peel potato, then cut into 1-inch pieces, and add to a small saucepan with 1 tablespoon salt and enough water to cover by ½ inch. Cover; bring to a boil. Uncover; cook until potatoes are tender, about 5 minutes. Reserve ¼ cup cooking water, then drain and return potatoes to pot. Mash with sour cream, reserved cooking water, and 1 tablespoon butter. Cover to keep warm.

Prep chicken: Pick leaves from half of the sage; reserve stems and thinly slice leaves. Halve fontina horizontally though the middle. Using a meat mallet or heavy skillet, pound chicken to an even ⅛-inch thickness; season with salt and pepper. Place cheese and sliced sage leaves on each breast; fold over to cover and pat to flatten. Sprinkle chicken with 1 tablespoon flour total.

Brown chicken: Heat 1 tablespoon oil in a medium skillet over medium-high. Add chicken and cook until lightly browned on one side, about 3 minutes. Flip chicken and transfer to same rimmed baking sheet. Save skillet for step 5. Bake on upper oven rack, until cheese is bubbling and chicken is cooked through, 6–8 minutes.

Make sauce: Heat 1 tablespoon oil, sage stems, and remaining chopped shallots in same skillet over medium. Cook, stirring, until shallots are softened, 1–2 minutes. Add broth concentrate, 2 teaspoons vinegar, and ¼ cup water. Bring to a simmer; cook until reduced by half, 1–2 minutes. Off heat, stir in 1 tablespoon butter. Season with salt and pepper.

Finish & serve: Add arugula and toasted walnuts to bowl with shallot vinaigrette, and toss to combine. Season to taste with salt and pepper. Remove and discard sage stems from sauce. Spoon sauce over fontina-sage stuffed chicken, and serve with mashed potatoes and salad alongside.