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Please don't boil the eggs too hard.
Literal
Egg [object-を] overly hard-ly boil-not-please.
あまり + a negative = 'not too X / not very X.' 固く is the adverbial form of 固い ('hard, firm') — note this is 固い (firmness of texture) rather than 硬い (rigidity), though the two are often interchangeable. ~ないでください is the standard polite negative request.