Hi MdAbuAsad!
There actually isn't anything technically wrong with the structure "that _____ that" because it isn't a structure, per se. What you have is a modifier describing another modifier (which happens all the time). That said, "that" noun modifiers can often be made a bit more concise by just omitting them (which is what the writers did in choice B).
The issue with answer A isn't the second that, it is actually the THIRD modifier (exploding) and the fact that the sentence doesn't have a verb for the relative pronoun clause "that a star..." (scientists found evidence that a star did WHAT?)
You could certainly rewrite answer choices A and B with or without "that" immediately following the word "star" and have them be correct and incorrect.
(A) ORIGINAL - wrong because the clause beginning "that a star..." doesn't have a verb.Astronomers have uncovered evidence that a star that was as bright as the full moon exploding into view 340.000 years ago, emitting dazzling radiation that could have disrupted Earth's protective ozone layer and sunburned our Stone Age ancestors.
Core: Astronomers have uncovered evidence that a star... (no verb, just lots of modifiers)
(A) Fixed - the clause beginning "that a star..." has a verb now (exploded).Astronomers have uncovered evidence that a star that was as bright as the full moon exploded into view 340.000 years ago, emitting dazzling radiation that could have disrupted Earth's protective ozone layer and sunburned our Stone Age ancestors.
Core: Astronomers have uncovered evidence that a star... exploded into view, emitting... radiation that could have disrupted...and sunburned.
(B) ORIGINAL - correctAstronomers have uncovered evidence that a star as bright as the full moon exploded into view 340,000 years ago, emitting dazzling radiation that could have disrupted Earth's protective ozone layer and sunburned our Stone Age ancestors.
Core: Astronomers have uncovered evidence that a star... exploded into view, emitting... radiation that could have disrupted...and sunburned.
(B) ORIGINAL - correctAstronomers have uncovered evidence that a star that was as bright as the full moon exploded into view 340,000 years ago, emitting dazzling radiation that could have disrupted Earth's protective ozone layer and sunburned our Stone Age ancestors.
Core: Astronomers have uncovered evidence that a star... exploded into view, emitting... radiation that could have disrupted...and sunburned.
Okay, but what about the "that" modifier. I can say either of the following and it be technically correct.
Her doctors discovered evidence that a treatment that was recently developed might cure her condition.
Her doctors discovered evidence that a recently developed treatment might cure her condition.
However, the second version is just more concise, and so the test (and good writers) are likely to omit "that" modifiers when possible for concision.
One final note - the modifier in choice (A) "that was as bright as the full moon" is potentially problematic because of the verb tense "was." The past implies that the star WAS as bright as the full moon but is no longer that bright. Presumably, the star is still around, so there is no reason to think we should be using the past tense. The version of the sentence without the "that" modifier doesn't require a verb within the modifier, and so avoids that issue.
Hope this helps!
Whit