I grew up drinking Turkish coffee, though we were only allowed to call it Greek coffee (long story...). Indeed, you are not supposed to drink it. Here is what you are supposed to do with that sludge:
1. Leave a tiny sip of coffee left, enough so that you can swirl the sludge around a bit
2. Slowly swirl the sludge around over the entire sides and bottom of your cup, then give it a couple of quick swirls at the end.
3. Turn it over quickly onto your saucer and leave it for a minute or two
4. Transfer upside down cup onto a napkin and leave it there for another minute or two
5. Hand cup over to old family friend who can tell fortunes from the patterns left by the coffee grounds
6. Watch her intently study the grounds, brows furrowed
7. If you are a single woman, the fortune will almost always include some prediction of marriage. Blush as everyone around you squees and giggles and makes comments about impending, though still imaginary, nuptials)
8. If you are a married woman, your fortune will almost always include a prediction of babies. Again, blush.
(7 and 8 are based only on my experience based on my Thea Nafsika's readings)
9. When she is finished, the coffee ground reader will fake spit into your cup and hand it back to you.
10. If you thank her, her predictions will not come true, so don't say "Thank you." Unless of course you don't want to get married and didn't like her prediction that you would be soon. Ignore your mother's scowl.
11. Go finish helping your mom lay out a huge spread of desserts.
That's how we did it. Some more info here with a pic of what the grounds look like for a reading.
Reading Coffee Cups