The first task, clearly, is to detach the sense of identity from the descriptions of yourself. This does not mean to find another description that would be the correct one. It means to realize there is no description of you.
Yes, awe arises during the extraordinary: when viewing the Grand Canyon, touching the hand of a rock star like Iggy Pop, or experiencing the sacred during meditation or prayer. More frequently, though, people report feeling awe in response to more mundane things: when seeing the leaves of a Gingko tree change from green to yellow, in beholding the night sky when camping near a river, in seeing a stranger give their food to a homeless person, in seeing their child laugh just like their brother.