Preheat your oven to 400°F. While it heats, trim the asparagus and cut into 1-inch pieces, then slice the green onions into thin rounds, separating white and green parts. Flake the precooked salmon into large chunks, keeping the pieces intact for better texture in the finished pie. Shred the Swiss cheese if not using pre-shredded, and measure out all dry and wet ingredients so they're ready to combine.
Lightly grease a 9-inch pie plate with butter or cooking spray. Distribute the asparagus pieces, white and light green parts of the green onions, flaked salmon, and 3/4 cup of the Swiss cheese evenly into the prepared pie plate. This layering approach ensures the vegetables and salmon are distributed throughout the pie rather than settling to the bottom. I like to keep the salmon chunks visible and intact during this step so they maintain their texture and don't break apart when you pour the batter.
In a medium bowl, whisk together the eggs and milk until well combined and smooth. Add the baking mix, basil, salt, and pepper, whisking until just combined—don't overmix, as some small lumps are fine and will cook out. Pour this custard mixture directly over the vegetables and salmon in the pie plate, letting it settle and distribute evenly. The batter will seep down between the ingredients and create a quiche-like texture as it bakes.
Place the pie plate in the preheated 400°F oven and bake for 30 to 35 minutes, until the custard is mostly set but still slightly jiggly in the very center. The top should be light golden brown. I find that checking at 30 minutes helps prevent overbaking, which can make the pie dry.
Remove the pie from the oven and quickly sprinkle the remaining 1 cup of Swiss cheese evenly over the top, including the darker green parts of the green onions if desired for color. Return to the oven and bake for an additional 2 minutes, just until the cheese melts and becomes slightly golden. This final step adds richness and a beautiful cheese top without allowing the custard to overcook.
Remove the pie from the oven and let it rest at room temperature for 5 minutes before slicing. This brief resting period allows the custard to set slightly more and makes serving easier without the pie falling apart.