To unlock A Rare Delicacy, you need to deliver one of the "rare" fish variants to any Hunter's Call trader at any seapost. Note that the "rare" fish are NOT the same as "trophy" fish, nor are they the "glowing" fish you get when you fish at night.
Luckily, rare fish are caught via the same method as their more bland counterparts, so you won't need any special bait or to be in any special location. You just need patience, and a bit of luck. (I had probably spent about an hour total time fishing before I caught my first rare spashtail, for what it's worth.)
Unfortunately, the game doesn't tell you when you've hooked a rare variant, so you'll probably need to consult a list like I did. Here are the rare variant colors of each fish type, plus where and how to catch them:
Splashtails: Umber (anywhere, no bait required)
Pondies: Bright (any freshwater, no bait required)
Islehoppers: Honey (near any island, no bait required)
Plentifins: Bonedust (Shores of Plenty, earthworms)
Wildspashes: Muddy (The Wilds, earthworms)
Ancientscales: Bone (Ancient Isles, leeches)
Devilfish: Forsaken (Devil's Roar, grubs)
Battlegills: Sand (skeleton forts or ships, grubs)
Wreckers: Sun (shipwrecks, earthworms)
Stormfish: Shadow (storms, leeches)
Hope that helps! If you need more info or have any general fishing questions, feel free to leave a comment or send me a PM and I'll be happy to assist.
Fish on!