When you do something composedly, you do it calmly and deliberately. You might composedly repair a broken window before your parents get home, instead of getting upset like your younger brother.
The adverb composedly is especially useful for describing something that's done in a way that displays common sense and a clear head. It implies calmness in the face of chaos, like when your aunt sits composedly with her cup of tea after firefighters have put out a fire in her kitchen. The root is the Old French word composer, "put together or arrange," from the Latin ponere, "to put or place."