I know that Harry and his friends eat meat but do they eat magical meat?
Are there edible magical creatures, or do they only eat non-magical meat?
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I don’t recall any instance in the main books where they ate anything from magical creatures (Voldemort’s unicorn snacks aside), but there are a few examples in the wider Potterverse.
We’ve seen several instances of people eating bits of dragon.
You can eat Flobberworms, although apparently people choose not to.
Again from the HP Wiki:
Flobberworm fritters are flobberworms fried in batter. This dish is served at Hogwarts School of Witchcraft and Wizardry, however the student body did not seem to care for it.
This apparently comes from the Philosopher’s Stone game, but I’m not digging up a clip.
You can eat (part of) a Murtlap.
Quoting from Fantastic Beasts and Where to Find Them:
The Murtlap is a rat-like creature […]. It has a growth upon its back resembling a sea anemone. When pickled and eaten, these Murtlap growths promote resistance to curses and jinxes, although an overdose may cause unsightly purple ear hair.
You can drink the blood of a Re’em.
Again quoting from Fantastic Beasts:
Re’em blood gives the drinker immense strength, although the difficulty in procuring it means that supplies are negligible and rarely for sale on the open market.
in HP: GOF Ron expresses his desire to eat off-menu. There's no special indication that he's joking:
‘Oh, hurry up,’ Ron moaned, beside Harry. ‘I could eat a Hippogriff.’
and Hagrid serves a casserole that contains a talon. There's no indication that this comes from a magical animal, though I'm at a loss to name any non-magical bird that tastes of beef.
They ended up having lunch with Hagrid, though they didn’t eat much – Hagrid had made what he said was a beef casserole, but after Hermione unearthed a large talon in hers, she, Harry and Ron rather lost their appetites
The other answers are excellent, but the first thing that came to my mind was the Lovegood family and how they love Freshwater Plimpy Soup and how it is claimed that others are always requesting the recipe. The exchange happens in The Deathly Hallows.
In lieu of an exact book quote (which I am unable to source for right now), here's a quote from the HP wiki:
When Harry Potter, Ron Weasley and Hermione Granger visited Xenophilius Lovegood in 1998, he covered the kidnapping of his daughter Luna by saying she was out fishing for freshwater plimpies near Bottom Bridge. Xenophilius later notes that people are always requesting the Lovegood's recipe for freshwater plimpy soup after he explains the Deathly Hallows to Harry, Ron, and Hermione.
Plimpies are definitely magical creatures, albeit not very "impressive" ones: http://harrypotter.wikia.com/wiki/Plimpy