The day before cooking, rinse fish inside and out and pat dry with paper towels. Make three or four parallel slashes about 1 in. apart in thickest section of each fish, halfway to bone, so it cooks evenly. Rub with salt (about 1 1/2 tbsp. per fish). Chill, uncovered, until ready to cook.
Meanwhile, prep artichokes (if using frozen artichoke hearts, skip to step 4). Fill a large bowl with water, squeeze in juice of 1 lemon, and add juiced peels. Working with 1 artichoke at a time, trim off and discard stem. Break off and discard outer leaves down to tender, pale green inner ones. Slice off pointy, darker green end and discard, then pare off dark green leaf bases. Cut artichoke in half lengthwise and scoop out fuzzy center and any thorny petals. Drop artichoke into lemon water.
Put a steamer basket in a large pot and fill with enough water to just touch bottom of basket. Cover and bring to a boil, then add artichokes to basket. Steam until tender enough to be easily pierced with a knife, 5 to 8 minutes. Rinse in a colander with cold water to stop cooking.
Preheat oven to 450°. Remove fish from refrigerator and pat dry inside and out. Lightly oil two large rimmed baking sheets and set a fish on each one. Rub fish all over with oil (about 1 tbsp. per fish). Mix 2 tbsp. minced oregano with 2 tbsp. salt; rub mix over fish and into slashes.
Slice 1 lemon crosswise and lay a half inside each fish, along with several branches of oregano. Slice remaining 3 lemons into quarters lengthwise, then slice each quarter in half lengthwise. Cut grapefruit the same way.
Put lemons and grapefruit in a bowl with artichokes, 1 1/2 tsp. minced oregano, 1 tsp. salt, 1/2 tsp. pepper, and 3 tbsp. remaining oil. Toss gently to coat.
Add artichokes and citrus to baking sheets on either side of fish, dividing evenly and spreading into a single layer. Roast fish until opaque down to bone (insert a knife in a slash to check), 20 to 25 minutes.
Meanwhile, make lemon-caper browned butter: In a small pan, melt butter over medium-high heat and cook until beginning to brown. Pour in cream and boil, whisking often, until separated and well browned, 10 to 15 minutes. Add capers and 1/2 tsp. each salt and pepper. Cook until capers start to crisp and open up into petals, about 1 minute. Whisk in 1 tbsp. lemon juice and 1 tsp. minced oregano. Pour into 2 small sauce bowls.
To serve, lift each fish and its artichokes and citrus to a platter. With a sharp knife, cut skin behind collar of fish, moving from top of fish to belly; then slice lengthwise just under top of fish. Peel back skin with two large serving spoons and transfer it to a plate. Set platters at either end of table and let everyone dig in themselves. Or, you can serve them: Spoon fish onto plates from top fillet, then lift out spine (set on plate with skin) and spoon out bottom fillet. Drizzle with lemon-caper browned butter and add some artichokes and citrus.
Make ahead: Steamed artichokes, up to 1 day. Lemon caper browned butter, up to 1 day (reheat on stovetop or in microwave).
Drink Pairing: A bright but round and aromatic white wine like Buty 2014 Se´millon/Sauvignon Blanc/Muscadelle (Columbia Valley; $25). Puckery citrus and resiny herbs are mellowed by honeyed pear and a rich mouth-feel.