I think its a head gasket. Running your
oil at the temps your running at is burning the water out of the
oil in the pan and the motor. The valve covers run quite a bit cooler and cool faster when you shut the motor down..hence the milk shake but not the creamy
oil.
Are you loosing water? Something you might try if you have it back together and didnt fix the leak. Put a radiator pressure tester on the radiator. Pump it up to about 15 psi...fire it up and let it run for just a few minutes. Then check your oil and see if you have any coolant present. Need to check it before it has a chance to get hot enough to burn it off.
Just a thought.