an utter disgrace - utter is an adjective modifying the noun, disgrace utterly disgraceful - utterly is an adverb modifying the adjective, disgraceful

One way you could think of this . . . . they are very similar to COMPLETE and COMPLETELY. So, if you know which of those would sound better, it will likely be the same for UTTER and UTTERLY. My cake was COMPLETELY/UTTERLY burnt. My cake was a COMPLETE/UTTER disaster. People sometimes even use the phrase COMPLETE AND UTTER for emphasis, even though it's redundant.