Slice tomatoes about 1/8 inch thick and lay on a tray lined with paper towels. Sprinkle with salt, stand for 10 minutes and blot both sides thoroughly.
Beat butter with basil, sherry vinegar and pepper until evenly mixed. Spread a thin layer over one face of all 12 bread slices.
Cover six buttered slices with one even layer of tomatoes, trimming pieces to fill gaps without building a thick stack. Close with the remaining bread, buttered faces inward.
Cover and refrigerate for 10 minutes to firm the butter. Trim crusts carefully with a serrated knife without pressing the tomatoes into the bread.
Cut each sandwich diagonally in both directions to make 24 small triangles. Serve two per portion, keeping the tray covered and refrigerated until serving the same day.