Preheat oven to 375°. Heat olive oil in a large, heavy ovenproof frying pan (not nonstick) over high heat until oil is hot but not smoking. Add chicken breasts, top side down, and cook until golden, 2 to 3 minutes. Turn chicken over.
Add shallots and garlic to pan with chicken, stirring occasionally so garlic doesn't burn, until shallots are soft and translucent, 3 to 4 minutes. Add chicken stock, tarragon sprigs, potatoes, 1/2 tsp. salt, and 1/4 tsp. pepper and cook until mixture begins to boil, about 1 minute.
Lay haricots verts over chicken, then cover pan and put in oven. Bake 14 to 16 minutes, or until chicken is just cooked through (cut to check) and haricots verts are tender. Transfer chicken to a plate.
Season haricots verts-potato-stock mixture with salt and pepper to taste. Divide vegetables among 4 rimmed plates or shallow pasta bowls, top each with a piece of chicken, and spoon sauce over all. Garnish with chopped tarragon and almonds.