Another personal opinion, but I don't think you need to worry about starting with a different question or anything like that, just treat the person like any other person you'd speak to. If I talk with someone that has a stutter, I usually just keep a neutral expression and wait for them to say what they want to say, even if it takes a few minutes. I definitely don't think it's a fail on your part, it sounds like you are way more aware than most people are about stuttering and doing the right kind of stuff, I gotta commend you!