Place tomato slices on several layers of paper towels and sprinkle with salt. Let sit for 15 minutes, then press with additional paper towels to extract excess moisture.
Using fork, mash 1/3 cup mayonnaise and blue cheese together in small bowl until smooth.
Spread blue cheese mixture on 8 slices toast. Spread 3 tablespoons mayonnaise on remaining 4 slices toast. Place 4 cheese-coated toasts on work surface. Layer each with sliced tomato, egg, and avocado. Place plain mayonnaise-coated toasts, mayonnaise side down, over avocado; spread tops of toast with remaining mayonnaise. Continue layering with 2 strips bacon and sliced chicken. Top with remaining blue cheese-coated toasts, facing down. Cut each sandwich into triangles. Serve.