Place artichokes, lemon juice, 2 tablespoons olive oil, salt, and fill pot with enough cold water to cover the artichokes in a large pot. Boil for 30 to 35 minutes or until tender. Remove artichokes from pot and let cool.
Preheat grill. When artichokes have cooled, slice in 1/2 lengthwise and remove the choke. Brush each half with the remaining oil, season with salt and pepper and grill cut-side down for 1 to 2 minutes. Turn over and grill an additional minute.
Place tomatoes, salt and pepper, vinegar, garlic and basil and olive oil in a blender and blend until smooth.
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