4. For the fish: Remove bones from cod with a pair of tweezers or pliers. Cut cod into four 3"-4" pieces (save remaining pieces for another use). Put 4 cups of the oil, garlic, and chiles into a medium pan and heat over medium-low heat until temperature reaches 175°. Add cod and poach until cooked through, 8-10 minutes. Meanwhile, heat remaining oil in a large skillet over medium heat. Add mushrooms and cook, stirring often, until just cooked through, 6-8 minutes. Add shallots and cook, stirring, until soft, 1-2 minutes. Season to taste with salt and pepper. Transfer cod with a slotted spatula to paper towels to let drain.