It is legal to drive without doors. As far as mirrors, I think in CT you have to have one rearward facing mirror, but that one I'm only 90% positive on, I have 2 mirrors (rear view and drivers side) anyway so I never paid much attention.
I was just at Troop H in Hartford doorless with my set up & the cops had no problem with that. They did have issue with my fenders though, my tires stick out past them. But he said he didn't care. I've had 2 cops now say it's illegal but that they don't care.
CT requires one rearward facing mirror. You will be just fine door-less.
OTOH, I HATE swiveling my neck around to the driver's side to change lanes. I solved that by getting a Harley long-stemmed mirror that I mount to the top hinge:
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Conn. Gen. Stat. ß 14-99. Mirror. Motor vehicles with commercial registration to allow others to pass. Each motor vehicle shall be equipped with a mirror attached to and so located and adjusted on such vehicle as to give the operator thereof a clear reflected view of the highway directly to the rear of or on a line parallel to the left side of the body of such motor vehicle. Any person operating a motor vehicle with a commercial registration so constructed or which may be so loaded that the operator is prevented from having a free and unobstructed view of the highway immediately to the rear and at the left side of the same shall, by means of such mirror, make frequent observations of the approach of vehicles from the rear. When operating at below the posted speed limits and when so approached or overtaken, the operator of such motor vehicle shall drive to the extreme right of the traveled way as promptly as safety will permit, giving the vehicle approaching from the rear opportunity to pass. Violation of any provision of this section shall be an infraction.