Preheat the oven to 350°F with a rack in the center position.
Season the salmon fillets with the salt and set aside.
Using a serrated knife, trim away the crust from the sides of the brioche loaf, then make four long lengthwise slices that are about ⅛ to ¼ inch thick. Lay a slice of brioche on a cutting board and place a salmon fillet lengthwise in the center; it should fall within the border of the crust. Use your knife to trim away the bread around the salmon. It’s best to hold the knife flat against the edges of the salmon and press straight down (using the same motion as with a cookie cutter) to get an even cutout around the fish. Repeat with the remaining fillets and brioche slices, setting the finished fillets brioche-side down on a plate or tray.
Heat a large oven-safe skillet over medium heat. Put in the oil and butter; when the butter begins to bubble and foam, gently place the salmon, brioche-side down, in the pan. Work in batches if you can’t fit all the fillets in the pan at once. Shake the pan a little to make sure the brioche is not sticking, then scatter the garlic and thyme sprigs around the fish. Cook, swirling the pan occasionally and basting the oil-butter mixture over the fish with a spoon until the bread is golden brown, about 1 minute.
Flip the fillets brioche-side up, transfer the pan to the center oven rack, and bake for 3 to 4 minutes, or until the salmon is cooked through but still has a touch of dark pink inside (about 145°F). Keep in mind, it will continue cooking a little even after you remove it from the oven!
Blot the fish with a paper towel and transfer to plates, brioche-side up. Serve immediately.