In a large metal bowl, combine the shallot, mustard, salt, and plenty of pepper. In a small saucepan, heat the smoked butter over medium heat, swirling the pan until the foam subsides and it starts to turn a nutty caramel brown. Before it can turn black, quickly whisk in the lime juice (it will sizzle madly). Whisk the hot butter mixture into the shallot/mustard mixture, then whisk in the tarragon. Set aside for up to 1 hour. If it sits any longer the butter may solidify. If it does, just warm very gently in a low oven before serving, until the mixture becomes liquid again.
Build a charcoal fire and prepare the grill for direct cooking over medium-high heat (400° to 500°). Working quickly, cut the cod filet into four equal pieces, and spread all sides generously with the mayonnaise. Season lightly with salt and pepper.
Brush the grilling grate clean. Grill the cod pieces over direct heat until opaque at the center, 6 to 8 minutes total, turning over once with a metal spatula. Toss the baby greens in the bowl with the browned butter vinaigrette and mound the juicy salad over the grilled fish.
To smoke butter, freeze 4 ounces of Challenge European Style Butter. Place unwrapped butter in a shallow metal roasting pan or mini-loaf pan and smoke at 225°F for 50 minutes. The butter will melt. You can re-chill or refreeze the butter if you don’t plan to use it right away.
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