Cook the gnocchi according to package directions; drain.
Meanwhile, in a large skillet, heat 2 tablespoons butter over medium-high heat. Add the onion and bell pepper and cook, stirring often, until softened, about 5 minutes. Add the chicken and cook, stirring occasionally, until just cooked through, about 5 minutes; season with salt and pepper.
In a small saucepan, melt the remaining 2 tablespoons butter over medium heat. Whisk in the flour and cook until toasted, about 3 minutes. Whisk in the half-and-half and chicken broth; bring to a boil. Lower the heat and simmer until thickened. Whisk into the chicken mixture, then stir in the gnocchi. Bring to a simmer, then season with salt and pepper and top with the scallions.