Preheat the oven to 400º F. Preheat the grill to medium high.

Toss the asparagus and tomatoes with 2 tablespoons of the olive oil, and season with salt and pepper. Place the asparagus and tomatoes on the grill and cook for 2 minutes on each side. Remove from the grill. Cut the asparagus into 1/2- inch pieces and coarsley chop the tomatoes.

Heat 1 tablespoon of the olive oil in a large skillet over medium heat. Add the onions, season with salt and pepper, and cook, stirring, for 1 minute. Add the garlic, asparagus, and tomatoes, and continue to cook for 1 minute. Add the green onions and crabmeat. Season with salt and pepper. Continue to cook for 2 minutes. Remove from the heat and set aside.

Season the trout with the remaining olive oil, and season with salt and pepper. Place the trout on a parchment-lined baking sheet. Bake for 10 to 12 minutes.

Combine the lemon juice and shallots in a a saucepan over medium heat, and season with salt and pepper. Bring the mixture to a boil, then reduce the heat to medium-low and simmer until the liquid reduces by half, about 3 minutes. Whisk in the butter, one cube at a time. Remove from the heat and strain the sauce through a fine-mesh strainer. To serve, ladle the sauce over the bottom of each plate. Spoon the relish in the center of the sauce. Lay the fish on top of the relish. Garnish with basil. Serve warm.