Bring the broth and 4 cups water to a simmer in a saucepan; keep warm.
Meanwhile, melt 4 tablespoons butter in a pot over medium-high heat. Add the onion; cook until translucent, about 4 minutes. Add the rice and thyme; cook, stirring, until the rice is glossy, about 1 minute. Add the wine and cook, stirring, until the liquid is absorbed. Add 1 teaspoon Kosher salt. Ladle in the hot broth, about 1/2 cup at a time, stirring constantly, allowing all of the liquid to be absorbed before adding more. Continue until the rice is just tender, 20 to 25 minutes.
Remove the thyme. Stir in the parmigiano, the remaining 1 tablespoon butter, 1/2 teaspoon Kosher salt and pepper to taste. Gently stir in the fontina and chicken. Reserve about 2 cups risotto for Risotto Cakes. Divide the rest among bowls; top with parsley and more fontina.