In heavy-bottomed saucepan, heat butter. Place mushrooms and carrot in pan and heat until softened, about 4 minutes. Add miso broth, peas, rosemary, and black pepper to pan; bring to a very gentle simmer and heat for 3 minutes. Dissolve cornstarch in 1/4 cup water and stir into pan; continue to heat until sauce has thickened. Stir in cooked chicken.