The brief
The A3 had a stack of symptoms going at once: bluish or grey smoke from the exhaust, oil disappearing faster than it should between changes, sluggish acceleration, a rough, uneven idle that occasionally threatened to stall, and the check engine light on. He brought it in. That combination points squarely at the oil separator in the crankcase breather. As the engine runs, oily mist builds up inside the crankcase, and the breather system vents that back into the intake to be burned cleanly, with the separator catching the oil so only the gases go through. When the separator fails, raw oil mist gets pulled straight into the intake and burned, that's the smoke and the oil consumption, and the failed unit upsets the crankcase pressure, which roughens the idle and trips the light. A failed oil separator doesn't recover, so it needs replacing.
The diagnosis
Diagnostics confirmed it, fault codes for the crankcase ventilation system and a lean mixture, and a check showed the separator wasn't holding pressure as it should, pulling oil through the breather into the intake. The intake, the injectors and the rest of the engine checked out, the oil burning was coming through the breather, not past the rings. That's a replacement. The separator and its diaphragm are sealed into the breather unit, you don't rebuild it on the car, so the call was a complete unit.
The work
The old oil separator was removed and a new genuine Audi-spec unit fitted with fresh seals, the breather hoses checked over while everything was apart. Then the fault codes were cleared and the engine's fuelling adaptations reset so it could relearn against a breather system that was working. A road test confirmed a steady idle, no smoke from the exhaust, the light staying off, and the response back.
The outcome
Steady idle, clean response, no smoke from the exhaust, no warning light, and the oil consumption back to normal. The A3 went home running properly again. A failed oil separator quietly feeds the engine its own oil and keeps the idle unhappy the longer it's left, so changing it and letting the engine relearn put the running right.