Very difficult. You would need a way for the Norman population to feel more "English" than "French". Normans settled in England after the conquest of 1066, rather than vice-versa. Even English rule ultimately collapsed in Gascony, where it was popular and lasted for at least 3 centuries. I can't see a permanent English presence in Normany.
At the time of the 11th and 12th centuries, just about every region in France was autonomous, and had a long-enough history of it. As for the Plantagenet realms, it was not so much an 'English' empire, but a collection of autonomous realms who were united insofar as they were ruled by the same dynasty. A dynasty from Anjou.
If you asked anyone back then how they would have identified themselves, it would have been with the duchy or county they were born in before anything else. Plus, many parts of France were divided by linguistic and ethnic lines to some degree or another. For example, Bretons did not view themselves as French, and nor did the French really count them among their own. And there were the Gascons, the Provencals and the Aquitainians, who had a long history of regional autonomy and cultural distinctiveness from the northern French. And there were even the French-speaking Burgundians.
England and its institutions were owned and operated mostly by Normans. By families whom owned land on both sides of the English Channel. From their perspective, the Normans saw the occupation of their duchy by Philip II of France as a conquest. Some of their noblemen may have switched allegiances from King John to Philip in 1202-4, but they still wished to live under their own laws. In the centuries after Philip's takeover of the duchy, the crown needed the co-operation of the Exchequer in Rouen to levy new taxes in Normandy. The duchy's formal existence was finally ended in 1469 by King Louis XI.