Are there any hangover cures proven to work?  If so, what are they?
There are no proven 'cures' for hangovers, but there are some remedies that may help alleviate hangover symptoms:

• Drink plenty of water to stay hydrated. Alcohol is dehydrating, so drinking water can help with headaches and fatigue.

• Get plenty of rest. Sleep helps your body recover from the effects of alcohol.

• Take an over-the-counter pain reliever like ibuprofen or acetaminophen to reduce headache and body aches. These can help reduce inflammation caused by drinking.

• Eat foods high in vitamin B and potassium like bananas, eggs, and nuts. Alcohol depletes these nutrients, so replenishing them may help improve energy and mood.

• Avoid caffeine which can dehydrate you further.

• Avoid greasy or spicy foods which can irritate your stomach.

• Time is the only true cure. Most hangover symptoms will subside as your body metabolizes the alcohol and regenerates nutrients, which typically takes 24 hours.

The truth is no remedy is scientifically proven to cure a hangover. The only sure way to prevent hangovers is to drink in moderation or not at all. But the strategies above may provide some relief from symptoms for many people.
Somebody on reddit told me that L-Theanine kinda helps with hangover by speeding up the processing of the alcohol. Is that true?
There is currently no scientific evidence to support the claim that L-theanine can help with hangovers by speeding up the processing of alcohol. L-theanine is an amino acid found in green tea, and while it may have some benefits for reducing stress and anxiety, there is no research indicating it has any effect on alcohol metabolism or hangover symptoms. It's important to be cautious of anecdotal claims about hangover remedies, and always consult with a medical professional if you have concerns about your health.