How to Make It
In a saucepan of boiling water, blanch the parsley until just limp, about 30 seconds. Drain, rinse and squeeze dry. Transfer the parsley to a blender, add the ice cubes and 1 tablespoon of olive oil and blend until smooth. Transfer to a bowl and whisk in the vinegar, lemon juice and sugar. Whisk in 3 tablespoons of the oil and season with salt and pepper.
Heat 1 tablespoon of the oil in a nonreactive skillet. Add the carrots and cook over moderately low heat, stirring, until soft and caramelized, about 20 minutes. Transfer to a bowl. Meanwhile, in a saucepan of boiling salted water, cook the peas until tender, about 3 minutes for frozen and 10 minutes for fresh; drain.
Heat 2 tablespoons of the olive oil in the skillet. Add the mushrooms and cook over moderately high heat, stirring, until tender, about 5 minutes. Add the peas, carrots, lemon juice, thyme and rosemary, season with salt and pepper and cook until just heated through, about 2 minutes; keep warm.
In a shallow bowl, combine the potato flakes and rice flour. In another shallow bowl, lightly beat the egg with 2 tablespoons of water. Season the cod fillets with salt and pepper. Dip the tops into the egg mixture, then press into the potato flakes. Put the remaining 1/4 cup oil in a large unheated nonstick skillet. Add the fish, potato sides down, and cook over high heat until golden, about 7 minutes. Carefully turn the fish, lower the heat to moderate and cook until the flesh flakes easily but is still slightly opaque in the center, about 3 minutes longer.
Spoon the vegetables onto 4 plates. Set the cod on the vegetables, crusted sides up, spoon the parsley vinaigrette around the fish and serve.