Preheat the oven to 325°F. Line 2 heavy large baking sheets with aluminum foil. Arrange the salami in a single layer over the baking sheets. Bake until the salami slices are amber brown, watching closely to ensure they brown evenly, about 10 minutes. Transfer the salami crisps to a paper towel-lined baking sheet to absorb the excess oil. Set aside to cool.
Do-Ahead Tip: The salami crisps can be made 8 hours ahead. Store airtight at room temperature.
Spoon a dollop of sour cream atop each salami crisp. Sprinkle the basil over and serve.