The phrase "without so much as flinching an eye" is used to describe someone who remains calm, composed, and unaffected by a challenging or difficult situation. Synonyms for this phrase include "unfazed," "stoic," "unperturbed," "resolute," "unruffled," and "undisturbed." These words convey a sense of strength and resilience, suggesting that the person in question is able to face adversity without losing their composure. Additionally, the phrase "with steely resolve" or "with nerves of steel" can be used to describe someone who remains resolute even in the face of insurmountable obstacles. These synonyms are useful for adding variation and richness to your writing, conveying the nuances of a character's behavior, mood, and emotional state.