Learn Turkish: Domatesleri soymadığını fark ettik. Yani, sosun kıvamı bozulacak. - We noticed that you didn't peel the tomatoes. Meaning, the sauce's consistency will be ruined.

Yani is used to draw a conclusion, state a consequence, or summarize what was just said (like saying "meaning..." or "so..."). Şöyle ki is used to elaborate or explain exactly how something is true (like saying "namely..." or "it's like this..."). If you used şöyle ki here, the listener would expect a detailed explanation of how you noticed the tomatoes, rather than the ruined sauce consequence.

Can we break down all the suffixes on soymadığını?

Yes, it breaks down as soy- (peel) + -ma- (not) + -dık (the participle turning the verb into a fact) + -ın (your) + -ı (accusative case). Literally, it means "your not-peeling." The final -ı is there because this whole word acts as the direct object of the verb fark ettik (we noticed).

Without a pronoun like senin, couldn't soymadığını also mean "that he/she didn't peel"?

Exactly! If the subject were "he/she", the word would be built as soy + ma + dık + ı (his/her) + n (buffer) + ı (accusative), resulting in the exact same spelling: soymadığını. Without pronouns like senin or onun in the sentence, you must rely entirely on context to know who forgot to peel the tomatoes.