In a food processor, combine the basil with the oil, tomatoes, nuts and garlic and process to a paste. Stir in the cheeses and season with salt and pepper.
In a bowl, blend the butter with the anchovy paste and refrigerate. In a small saucepan, boil the demiglace with the herbes de Provence until reduced to › cup, 3 minutes. Strain the sauce into a clean saucepan and stir in the vinegar. Bring the sauce to a simmer. Off the heat, whisk in the anchovy butter, 1 tablespoon at a time. Season with salt and pepper, cover and keep warm.
Light a grill or heat a grill pan. Oil the fish, season with salt and pepper and grill over high heat until browned on the outside and rare within, 1 minute per side.
Season the beef with salt and pepper and sprinkle with the herbes de Provence. Heat a skillet. (If not using Kobe or wagyu beef, heat 2 teaspoons of vegetable oil until shimmering.) Add the meat and cook over high heat until browned outside and medium-rare within, 2 minutes per side.
In a bowl, toss the sea beans with the olive oil and lemon juice; season with pepper.
Place dots of pesto on each plate and top with rounds of fried Asian eggplant. Arrange the fish and beef on the plates and garnish with the sea beans. Spoon the anchovy butter sauce over the beef and serve.
Sea beans are intensely salty sea vegetables. They are available at specialty-food markets or from sidwainer.com.
For wine, Aldo Sohm's pick is an equally luxurious red Bordeaux.
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