Place an oven rack in the center of the oven. Preheat the oven to 400 degrees F. Line a baking sheet with parchment paper. Set aside.
In a medium bowl, combine the flour, cornmeal, salt, baking powder, and baking soda. Add the butter. Using fingertips or a pastry blender, work the butter into the flour until the mixture resembles a coarse meal. Stir in the Parmesan and green onions. Add the buttermilk and olive oil. Stir until the mixture forms a dough. Drop the dough onto the prepared baking sheet in 12 equal-sized pieces. Bake for 18 to 20 minutes until golden. Cool the biscuits for 10 minutes then transfer to a rack to cool completely.
Slice each biscuit in half and spread the bottom half with 1 teaspoon of lemon butter. Add some arugula leaves and place a small slice of smoked salmon on top. Add the top half of the biscuit and serve.
In a small bowl, mash the butter and lemon zest together using a fork. Refrigerate until ready to use.