I don't know of a major philosopher who has analysed embarrassment at length as as a focal concept, but Purshouse provides a useful short analysis. You will find, I think, that a fair amount of conceptual analysis of embarrassment is done in books that are classified as psychological but also treat the subject philosophically. You might try :
Luke Purshouse, 'Embarrassment: A Philosophical Analysis', Philosophy, Vol. 76, No. 298 (Oct., 2001), pp. 515-540. (Philosophy)
B. Szabados, 'Embarrassment and Self-Esteem', Journal of
Philosophical Research 15 (1990), 341-9. (Philosophy)
Dacher Keltner and Cameron Anderson, 'Saving Face for Darwin: The Functions and Uses of Embarrassment', Current Directions in Psychological Science, Vol. 9, No. 6 (Dec., 2000), pp. 187-192. (Not philosophy but with philosophical bearings)
W. Ray Crozier, ed., Shyness and Embarrassment, ISBN 10: 0521172195 / ISBN 13: 9780521172196. (Not philosophy but with philosophical bearings)
Christopher Ricks, Keats and Embarrassment, ISBN 10: 0198128290 / ISBN 13: 9780198128298. (Not philosophy but with philosophical bearings)
Robert J. Edelmann, The Psychology of Embarrassment, ISBN 10: 0471914290 / ISBN 13: 9780471914297
Published by Wiley, 1987. (Not philosophy but with philosophical bearings)
Tangney, June Price, Self-Conscious Emotions: The Psychology of Shame, Guilt, Embarrassment, and Pride, ISBN 10: 0898622646 / ISBN 13: 9780898622645
Published by The Guilford Press, 1995. (Not philosophy but with philosophical bearings)