Nowhere have I been able to find why the hell this is a thing.

As an edit: 108,001 people reported to have side effects when taking Hydrochlorothiazide. Among them, 25 people (0.02%) have Breast discharge

Goljan had a lecture that mentioned that "If a patient has galactorrhea, review every drug they're taking since many drugs cause galactorrhea."

The only thing of possible relevance in this Q-stem is that she takes a medication, therefore the answer of "drug effect" is the most likely reason for her galactorrhea.

" Other classes of medications that cause hyperprolactinemia include antidepressants, antihypertensive agents, and drugs that increase bowel motility. Hyperprolactinemia caused by medications is commonly symptomatic, causing galactorrhea, menstrual disturbance, and impotence. It is Important to ensure that hyperprolactinemia in an Individual patient is due to medication and not to a structural lesion in the hypothalamic/pituitary area; this can be accomplished by (1) stopping the medication temporarily to determine whether prolactin levels return to normal, (2) switching to a medication that does not cause hyperprolactinemia "

https://www.ncbi.nlm.nih.gov/pubmed/16092584

"Non-dose-dependent side effects — Although low-dose therapy seems to minimize the metabolic complications induced by a thiazide or thiazide-like diuretic, it may not necessarily eliminate other side effects. As an example, as many as 25 percent of men treated with 25 mg/day of chlorthalidone develop a decline in sexual function [34]. Sleep disturbances can also occur, particularly if the patient is on a low-sodium diet [34]. How these problems occur is not known."

https://www.uptodate.com/contents/use-of-thiazide-diuretics-in-patients-with-primary-essential-hypertension#H2250530998

Possible explanation:

If the pt is taking a thiazide (which is K depleting), it might have also been given with a K sparing drug such as spirinolactone.

Spirinolactone has endocrinologic effects such as gynecomastia and galactorrhea.