Oil leak?

I will start at the beginning. If that amount of oil (a gallon) is actually being burned, you would expect to see the tailpipe smoking in your rearview mirror.

I accidentally left an oil dipstick raised on another VAG car and was amazed at how much oil had come out. BUT it was all sitting in the engine tray! You have to remove it and look! At the time I couldn't reconcile how much oil had come out of that small dipstick hole. PS it does have an O ring seal.

What I am saying is a gallon oil is either being burned or lost down to the engine tray. You have to remove the tray and check it for oil leaks. Those trays have molded 'pockets' which can hold quite a lot of oil, but eventually it should end up on your driveway!

If a lot of oil was being burned it would show up on all or some of the spark plugs (assuming a gas engine) which would be very black and coked. You need to take them out and look.

Another issue could be incorrect oil level when first filled. When oil is drained and clean oil added, it is very hard to find the oil level on the dipstick. You can get a black smear of oil making it hard to find the true level when it could be below the minimum mark. I paint my dipstick with white correction fluid and re-check the level after the car has rested 1/2 an hour. V.W dipsticks can be notoriously hard to read. But once the oil level gets below the minimum mark, this usually triggers the low oil level warning. Unfortunately, trainees are often given the job of changing oil and can get it wrong. If I drain oil out of my EOS I expect to get about a gallon out and to put back a full 4.5-5l bottle. If only half goes back I know something is wrong with my reading of the dipstick. In fact it is usually easier to misread and over fill than under fill.

The photos you post of the closed ventilation hoses don't prove to me that's where a gallon of oil is going?

I've added to this thread because of a recent experience which wasn't covered in previous replies. These engines use a closed crank case ventilation system. All fumes (and pressure) are recirculated back to the inlet manifold. To allow the engine to 'Breath' they fit a PCV valve which is a cheap flapper valve. The PCV is a critical part ensuring near zero pressure inside the engine when running. If it's faulty, engine pressure will force oil out very quickly. Since the oil cap is a good seal, oil tends to come out via the dipstick port. If the dipstick is a good seal then oil under pressure will find other ways out such as timing cover gaskets. You find oil around a timing cover, engine crank seal or other gasketted cover and think that's the cause, when it's the consequence of high crank case pressure caused by a faulty PCV. If you spot the symptoms do the tests and confirm a PCV fault you can get lucky and stop leaks. Whilst the engine has been running with high crank case pressure, cover gasket sealing could have broken and some have had crank seals turn inside out! If the PCV was confirmed faulty then expect to have any leaking cover gaskets still leaking oil replaced or re-sealed. Many car engine covers are sealed using silicone gasket goo which will break easily under pressure.

Simple check: Idle the engine, pull the dipstick up and if oil shoots up reaching the hood, the PCV is faulty. If not fixed excess engine pressure can blow out crank shaft seals.