In a baking dish large enough to hold the salmon, combine the 1 cup oil and basil, vinegar, salt and pepper. Add salmon and turn to coat evenly with marinade. Cover with plastic wrap.
Refrigerate for at least 30 minutes or up to several hours.
Preheat the oven to 350*F. In a large, heavy skillet, heat 2 tablespoons of oil over medium-high heat. Sear the salmon fillets for 1 minute per side, then transfer to a large oven-safe dish. Bake about 10 minutes (if you want the center translucent, cook a minute or two less).
If serving chilled, let cool, then cover with plastic wrap and refrigerate.
Make the mayonnaise:
In a food processor, blend the herbs, shallots, lemon juice, oil, salt and pepper for 1 minute. Add the mayonnaise and blend well. Taste and adjust seasoning. If sauce is too thick, add a little more oil. Cover with plastic wrap and refrigerate.
Arrange the lettuce or spinach and the radishes on a large platter. Place the salmon on the greens, drizzle with the herb mayonnaise, and garnish with fresh minced chives. Serve with more sauce on the side. Enjoy!