Sneaking something with 目を盗む
人に見つからないように、こっそりする。
Do it secretly, so that people don't find out.
Goo Dictionary
Written as 〜の目を盗んで, this expression means to do something stealthily or in secret. Something is being done without being seen by someone or without their knowledge, with the hope of not being found out by the person identified directly before the の.
In English, a similar phrase is to do something “behind someone’s back”. In some similar cases, you may “slip past” someone.
