In a manner that shows appraisal; while evaluating or assessing something.
From appraising plus -ly suffix, converting the participle to an adverb.
When someone looks at you 'appraisingly,' they're doing it thoughtfully, as if measuring your value or worth. It's how a coach watches a new player, or a mentor observes a protégé—careful, evaluative, calculating.
Complete word intelligence in one call. Free tier — 50 lookups/day.