In a deep 12-inch skillet bring broth or water to a boil and add chicken breasts in one layer.
Reduce heat and poach chicken at a bare simmer, turning once, 7 minutes.
Remove skillet from heat and cool chicken in liquid 20 minutes.
discard liquid and cut chicken into 1/2-inch cubes.
Cut potatoes into 1/2-inch cubes.
In a large saucepan of boiling salted water cook potatoes until tender, about 15 minutes, and drain in a colander.
In a 12-inch heavy skillet cook onion, bell peppers, and garlic in butter over moderately high heat, stirring frequently, until softened, about 3 minutes.
Add spices, thyme, and salt and cook, stirring, 1 minute.
Add chicken, potatoes, scallion, parsley, and half-and-half and cook over moderately high heat, stirring occasionally, until half-and-half is reduced slightly and hash is thickened.