Are phrases beginning with "not that..." independent?

Which out of these two examples is the best way to punctuate the following? My confusion has arisen from not knowing if "not that I minded" counts as an independent clause.

"I used to have to walk down the entire country lane to get back from school, not that I minded. It was far more pleasant than having to walk down the busy roads."

"I used to have to walk down the entire country lane to get back from school. Not that I minded; it was far more pleasant than having to walk down the busy roads."


If it is independent, the second would be correct. If not, then it would be the first one.