In a small saucepan, combine the milk, water, and butter. Warm over medium heat until butter is melted. Cool for about 5 minutes, or until temperature is 120-130° F.
In the bowl of a stand mixer fitted with a dough hook (or in a large bowl if planning to mix by hand), combine 1 cup of the bread flour, the yeast, sugar and salt. Add the warm water mixture, followed by the beaten egg, mixing on low speed until incorporated.
Add the remaining bread flour 2 tablespoons at a time while mixing on medium low speed until dough almost clears the bowl and is only a slightly sticky to the touch. Knead until smooth and elastic, about 5-7 minutes more.
Gather dough into a ball and place in a greased bowl. Cover with plastic wrap and put in a warm place and allow to rise until doubled, 1 to 2 hours.
Punch down dough and let rest for a few minutes. Divide dough into eight equal pieces and shape into balls. Place on a sheet pan lined with parchment paper. Spray some plastic wrap with non-stick cooking spray and place over the shaped buns. Return to a warm place and continue to rise until doubled again, about 30 minutes.
When ready to bake, preheat oven to 400° F. Brush buns with egg wash. Bake until golden brown, about 10 minutes. Cool completely on a wire rack.
To make the salmon cakes:
Place the salmon in a medium skillet and add water. Sprinkle salmon with salt and pepper. Bring to a boil, then cover and simmer for about 15 minutes until just cooked all the way through. Let cool slightly. Use a fork to gently flake the salmon into a medium bowl.
Heat 1 tablespoon of the oil in a skillet over medium heat. Add the onions and cook until tender and translucent, about 5 minutes. Add onions to the salmon, along with 1 tablespoon olive oil, the cilantro, basil, salt, lemon zest, liquid smoke, egg and bread crumbs. Mix until just combined.
Heat the remaining 2 tablespoons olive oil in the skillet over medium high heat. Spoon the salmon mixture in mounds into the hot skillet (an ice cream scoops works great) and flatten into disks. Cook until brown, about 3-4 minutes, then flip and cook on the other side. Transfer to a paper towel lined plate.
To assemble the sliders:
In a bowl whisk together the mayonnaise, lemon juice, lemon zest, basil, cilantro and salt.
Spread the sauce onto the buns, then top with a salmon cake, lettuce and tomatoes. Serve immediately.
Note: Salmon cakes can be made up to 3 days ahead. Reheat in a greased skillet.