I learned that in Japanese you can put a sentence in front of a noun without の: like "買った本" or "もらったプレゼント".
In a JLPT N3 sample question I encountered "オープンしたばかりのレストラン". Could I omit the の and just say "オープンしたばかりレストラン"?
When you're modifying a noun with a verb phrase (e.g., 美智子が買った本), no particle is used between the verb phrase and the modified noun. However, ばかり acts like a noun, and when you're modifying a noun with a noun phrase in Japanese, the の particle is required. If it's helpful, think of オープンしたばかりのレストラン as the same construction as:
Whether ばかり is actually a noun is an interesting question. (It seems to me that it is, but I'm not sure.)