Good thing you guys added the honorifics! It's a good way for lay people to understand a bit about the language and culture of the original work.
Also, Millie definitely tried to cook and something went wrong. She could have cut herself fr but it's definitely something silly like "Oh no help me I can't cut this" (making some drama to get her crush to worry)
Looking at the way the blood is distributed, the cut definitely doesn't come from her wrists, it's most likely the palm of her hand or her thumb, which is a common accident when people with no skill have a sudden burst of confidence to feel like a Master Chef
Ofc the cliffhanger is bait and it's open for discussion if this type of bait that is used A LOT OF TIMES in media is really that funny (it isn't), but Millie won't try anything harsh. She knows the best way to go against her negligent mom is to get a cute gf and be happy.