Daniel Dae Kim was wide awake and scrolling through Twitter at 4 a.m. when he realized “The Good Doctor” was going to be a hit.

It was a Tuesday in September, and Mr. Kim, a 49-year-old actor known for his work on “Hawaii Five-0” and “Lost,” found himself in bed in western Bulgaria not far from the set of “Hellboy,” a movie he was filming for release next year.

He had a special interest in “The Good Doctor,” because it represented his first outing in a new role: executive producer. And so, although he had to go to work in just a few hours, he could not keep himself from checking the effusive social-media reaction to the show’s premiere episode on ABC.

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