Is it permissible to put Soup Mandels in hot (Yad Soledes) soup on Shabbos, assuming the soup bowl was filled in a standard way - pot taken off of fire, soup ladle inserted into pot and soup removed and put into the bowl in which it is served.
Yes. You can even put it in a cli rishon that's off the fire.
שקדיות מרק ... הנמכרות כשהן מוכנות לאכילה, מותר לתת אפילו בכלי ראשון, אם אינו עומד על גבי האש...
Soup mandels ... that are sold ready to eat, it is permissible to place them even in a cli rishon, if it is not on the fire. (Translation mine)
Source: Shmirat Shabbat C'Hilchata, 1:61
HaRav Chaim Bleier shlita in his Chukei Chaim series here (on page 2) addresses this question.
He writes as follows:
Adding Croutons or Soup Nuts to Soup on Shabbos
Croutons and noodles. Deep-fried soup croutons, which are considered cooked1, may be added to soup on Shabbos, even if it is yad soledes bo (מאור השבת ח"ב סי' ח' סק"ה) since we pasken ‘ein bishul achar bishul’ with dry foods (שו"ע סי' שי"ח ס"ד) The same applies to noodles....
Soup nuts. Although soup nuts are also deep-fried and have the same status (מו"ר בשו"ת שבט הקהתי ח"א סי' ק"ט), some are uncertain about them since the oil stays on their surface and does not permeate them; perhaps they should be considered baked, not entirely cooked, and adding them to hot soup would involve an issur of bishul achar afiyoh, which is a matter of dispute (ב' דיעות בשו"ע סי' שי"ח ס"ה) and should be avoided lechatchilo (רמ"א שם). Therefore it is best to put them into a kli shlishi, not a kli sheini (שו"ת שבט הלוי ח"ז סי"ב). Nevertheless, when considering that soup transferred from a pot to a bowl with a ladle may be a kli shlishi (מ"ב סי' שי"ח סקפ"ז), one may rely on this and be meikel for bishul achar afiyoh (מ"ב שם סקמ"ה).
1 Store-bought croutons made in a factory by deep-frying small pieces of bread have the status of less than a kezayis of cooked bread and are mezonos (שו"ע סי' קס"ח ס"י, הגרשז"א שש"כ פ"א סס"א). Even if they were fried in a bit of oil, not deep-fried, they are mezonos since they were made with bread baked specifically for croutons (מדריך הכשרות עדה"ח תשע"ז עמ' 162).
When the pot is taken off the fire, it is still a kli rishon. The soup ladel is a kli sheni and the bowl a kli shlishi.
- If the soup ladel gets as hot as the soup it may be a kli rishon as well.
- Not everybody holds of a kli shlishi.
- If the soup has a davar gush in it, e.g. noodles or kreplach, the davar gush may be treated like a kli rishon. It is unclear if that changes the status of the soup.
- If frying is like cooking, the rule ain bishul achar bishul should apply.
- If frying is like baking, it is a machlokes whether there is bishul achar afiyah.
As such, it depends on what you hold and how machmir you are on Shabbos. Either way you have who to rely on. IIUC, most people are lenient.