Oh, I adore this! Love Poppy's many, insightful thoughts on her profession, and this meeting of the minds over scientific problem-solving. The many potions references are fascinating; they show that this is an author who knows her/his stuff, and they feel perfectly natural as parts of the story that give it texture and characterise both Poppy and Severus. I love this kind of magical/scientific background!
Also, so many beautiful moments. I especially liked how Poppy barged in without a greeting and Severus didn't notice because his mind, too, was set on the matter, not politeness. And that you worked in references to his premature potions genius and complemented that by Poppy's knowledge in occupational health & safety (not that she couldn't obviously also still hold her own with a cauldron :))
The idea of the Phoenix tears as a possible solution seems so plausible, and hooray for inadvertently-matchmaking effects. The ending is just perfect. A gift between friends indeed.