Form the LLC in NH because that is where you are doing business. If you form the LLC in another state, you will have to register it in NH, anyway, so you will incur duplicate startup and ongoing costs.
Disclaimer: This information does not constitute legal advice and does not establish an attorney-client relationship.
Well there are two considerations.
First, where your business is based out of. For instance because of various organizational benefits a lot of companies register in Delaware. The founder might in fact never go to Delaware but that is the official "base".
And second, where your business is doing business out of. I am not a lawyer but in order to do business out of states you need to register as an entity if you are operating under a name other than yourself and have employees. If you are employing people in NH then I am 99% sure you need to register there.
So to your specifics if you aren't doing any business in CT you don't need to register there. You first need to figure out where you want your business to be based out of. Say Delaware? Then register there and register in NH as a foreign corporation. If your business won't benefit from registering in Delaware just register in NH. If you're doing business out of CT then maybe based your business out of there and register as a foreign corporation in NH.
I hope that helps.