Season the skate on both sides with salt and pepper. Place the flour on a large plate and season with salt and pepper. Dredge the skate on both sides in the flour and tap off any excess.
Heat 2 tablespoons of olive oil in a large nonstick skillet over high heat. Cook the skate until lightly golden brown and just cooked through, about 3 minutes per side. Transfer to plates or a platter.
Add the wine to the skillet and reduce by half. Add the cold Smoked Chile Butter, a few tablespoons at a time, and whisk until emulsified. Add the capers, lemon juice and tarragon and season with salt and pepper. Spoon the sauce over the fish and top with the tomato salad.
