Q) I fried onions in a parev frying pan, and accidentally mixed the onions with a milky eino ben yomo Have the onions become milky?
A) [An eino ben yomo utensil is a utensil that has not been used during the last 24 hour period for hot food[1]. The halachic ramification of such a utensil is that the “taste” contained within the utensil has become “pogum” (unfavourable) and consequently cannot usually affect other food.]
Even though an eino ben yomo utensil cannot generally impart a beneficial flavour and will therefore not forbid food that it is used with, when an eino ben yomo utensil is used in conjunction with a dovor chorif, the dovor chorif has the ability to rejuvenate the pogum taste contained in the eino ben yomo utensil and the dovor chorif will consequently be affected by the taste contained in the eino ben yomo utensil (second opinion quoted in Shulchan Aruch YD 96:1; Remo YD 96:3; Shach 96:6).
Therefore, the default position in your scenario is that the onions – which are a sharp food item – cause the eino ben yomo bliyos in the ladle to be rejuvenated; and the onions will therefore absorb these bliyos and become milky.
It should also be mentioned that the above is correct only if the onions were mixed with the ladle immediately after they started cooking. If the onions were already cooked by the time the ladle was inserted then there may be room for leniency and a sheiloh should be asked. This is because onions lose their sharpness during the cooking process, and by the time they are cooked they are no longer classified a dovor chorif. (Although it is difficult to quantify the exact stage of cooking at which the onions lose their sharpness.)
[1] וגם לא נשרה בתוכו משקה למשך כ”ד שעות – שאל”כ כבוש כמבושל