I remember that well. I presume you're referring to an episode of "The Outer Limits," which was based on a short story by Larry Niven. Having looked into the idea back when I read the story in the 1990s, I can say it's plausible in the sense that it's not strictly prohibited by any law of physics.
That said, a solar flare would have to be of precisely the right size, coming in at precisely the right trajectory for precisely the right duration, and with the moon in precisely the right spot for the scenario in the story to play out. More likely, the entire planet would be completely roasted, and all higher life forms extinguished.
Of course, most solar flares are not nearly that powerful. Typically, we experience transient disturbances in global telecommunications, spectacular aurorae, and little else when a flare hits. In the late 1800s a more powerful flare hit, and some people speculate that a flare that size could affect not only telecommunications but also power grids. However, the planet would not even be singed.
There are in fact several ways to burn the Earth, as you put it. Astronomer Phil Platt describes these in wonderful detail in his book Death from the Skies. In addition to a solar flare, a gamma ray burst from a nearby supernova (< 20 ly) would do it. I won't mention the others since I don't want to spoil the fun of the book.