I had eaten lobster rolls before, but had no idea until moving to the Northeast that there were so many variations.

When we moved up here, B learned about lobster rolls by passing by a McDonalds, which sells it on their menu. B immediately wanted to try it, but I insisted that he had never had a lobster roll before, and his first lobster roll should be a legit one! (nothing against McDonald's)

B grudgingly agreed to save himself for a real lobster roll for his first time, jokingly calling himself a "lobster roll virgin."
Fascinatingly, the town right next to me Mildford, claims to be the birthplace of lobster rolls!

This fact is actually backed up by the Encyclopedia of American Food and Drink, starting from the 1920s. Between pizza, hamburgers, and lobster rolls, Connecticut is quite the food innovator (source).
There are distinct types of lobster rolls - New England (Maine) vs Connecticut style. New England style (source, source, source)

New England (Maine) style
- served cold
- filling includes lobster, mayonnaise, tarragon and celery or scallions 
- served on a “New England” or “Frankfurter” roll (hot dog bun with flatter sides, toasted/grilled) 
- probably the more comment version you find in the US

Connecticut style
- served warm
- no mayonnaise, but lots of butter to dress the lobster roll
- also served on a split top bun

For me, hands down the Connecticut lobster roll is the way to go. I love my sandwiches hot/toasted, and I also generally dislike mayo. B is much less discriminatory and likes all lobster roll types.
After waiting almost a year, the first place B ate his lobster roll was a worthy one - Lobster Landing in Clinton Crossing. The quaint shack and outdoor eating surrounded by boats is picturesque, the sandwich size is generous with a good lobster meat ratio, and, most importantly, their lobster rolls are just amazing (the best I've had so far). It's highly reviewed in Tripadvisor, Yelp, and Google unsurprisingly.

We enjoyed the lobster rolls after a great day at the beautiful Hammonasset Beach, one of the prettiest  beaches / state parks near New Haven. Lobster Landing is just a short drive away.
B agreed that it was worth saving his "lobster roll virginity" for the best lobster roll in Connecticut, but he is also glad to be able to eat lobster rolls indiscriminately now without me bugging him. When my brother came to visit, they of course both tried the McDonald's roll (which is a much more cheaper way to try lobster rolls for the budget conscious).

Butter Herb Lobster Roll

Ingredients
1 lb lobster
4 tablespoons unsalted butter, softened
olive oil
garlic, minced
1 lemon, juiced and zested
parsley
fresh dill
salt and pepper, to taste
1 baguette
1 teaspoon chopped scallions, to garnish

Directions
Remove lobster meat from the shell and set aside (my meat was bought already removed from shell)

In a large saute pan, combine the softened butter, olive oil, garlic, lemon juice and zest over medium heat and melt. Add the lobster and coat until the lobster is warm, about 4-5 minutes. Add parsley, dill, salt and pepper to taste.

Cut the baguette into desired side. With a knife, split from top where you can add the lobster. Toast the baguette until golden brown.

Divide the lobster between the baguette and serve with a sprinkling of herbs on top. Serve warm. Enjoy!