Awkwardness is due to lack of social graces, and communication skills. Perhaps one can train on these in the absence of other, live humans--read, for example, "How to Work a Room," or "The Fine Art of Smalltalk"--but ultimately, one must learn to navigate awkwardness.
In fact, I might go so far as to argue that the "connection" you're talking about is generated only by _resolving awkwardness_. Where there's none to resolve, there's no connection to be made. Relationships worn smooth by years of resolution aren't valuable in the same way.
You can effortfully apply all the social graces and make correct small talk, and still come away from an interaction thinking, “that was excruciating.” These situations can be even more awkward than when someone actually commits a faux pas.
In fact, I might go so far as to argue that the "connection" you're talking about is generated only by _resolving awkwardness_. Where there's none to resolve, there's no connection to be made. Relationships worn smooth by years of resolution aren't valuable in the same way.