Adjust oven racks to middle & lower positions & preheat oven to 250 degrees F. Pat fish dry with paper towels & season each fillet with 1/4 tsp salt. Let sit at room temperature for 20 minutes.
2Meanwhile, toss artichokes & cornstarch in bowl to coat. Heat 1/2 C oil in 10" nonstick skillet over medium heat until shimmering. Shake excess cornstarch from artichokes & add to skillet; cook, stirring occasionally, until crisp & golden, 2 - 4 minutes. Add garlic & continue to cook until garlic is golden, 1 minute. Strain oil through fine mesh strainer into bowl. Transfer artichokes & garlic to ovenproof paper towel lined plate & season with salt. Do not wash strainer.
3Return strained oil to skillet & add remaining 1/4 C oil. Place onion 1/2 in center of pan. Let oil cool until it registers about 180 degrees, 5 - 8 minutes. Arrange fish, skinned side up, around onion (oil should come roughly 1/2-way up fillets). Spoon a little oil over each fillet, cover skillet, transfer to middle rack & cook for 15 minutes.
4Remove skillet from oven. Using 2 spatulas, carefully flip fish. Cover skillet, return to middle rack & place plate with artichoke/garlic mixture on lower middle rack. Continue to cook fish for another 9 - 14 minutes longer or until it registers 130 - 135 degrees F. Gently trasfer fish to a serving platter, reserving 1/2 C oil & tent fish loosely with aluminum foil. Turn off oven, leaving plate of artichokes in oven.
Process whole cherry tomatoes, shallot, vinegar, 3/4 tsp salt & pepper with reserved 1/2 C fish cooking oil in blender until smooth, 1 - 2 minutes. Add any accumulated fish juice from platter, season with salt to taste & blend for 10 seconds. Strain sauce through fine mesh strainer, pressing on solids to extract as mush liquid as possible; discard solids.
6To serve, pour vinaigrette around fish. Garnish each fillet with warmed crisped artichokes & garlic, parsley & tomato rounds. Serve immediately.
WHY POACH IN OIL VS OTHER LIQUID? Poaching in oil allows fish to retain more of it's juices than poaching in wine or broth, leading to remarkably moist, velvety results. This is because cooking in oil in inherently more gentle than cooking in water (see more notes). And while you might expect that fish poached in fat would be greasy, it actually absorbs very little oil. Why? In order for oil to penetrate the fish, moisture must exit first. But because oil & water repel each other, it's very difficult for moisture inside the fish to readily enter the oil. Hence, more of the juices stay in the fish. In fact, oil poached fish lost just 14% of it's weight during cooking, while water poached fillets lost 24%.
WHY FOOD COOKS SLOWER IN OIL THAN IN WATER...Even when both liquids are exactly the same temperature. This is true, it turns out, not just for fish but for any other food including eggs. But how can this be? Isn't temperature what determines speed of cooking? As it happens, equally critical is the liquid's thermal capacity, or how much energy is needed to change its temperature by 1 degree centigrade. Oil has roughly half the thermal capacity of water, which means it requires half the amount of energy to reach the same temperature as an equal volume of water. This, in turn, means it has less energy to transfer to food & will cook more slowly.