Preheat the oven to 400°F. Optionally, line a couple of baking sheets with parchment paper (this helps prevent sticking). Lightly oil the parchment paper (or the baking sheets, if not using parchment paper).
Place the potatoes in a large pot and cover them with salted water (Note 1). Place the pot over high heat and bring the water to a boil. Boil the potatoes until they're tender enough to be easily pierced with a fork, 15 to 20 minutes. Drain them into a colander when done.
Let them sit for about 10 minutes to steam dry, which will help them crisp up in the oven.
While the potatoes boil, stir 3 tablespoons of olive oil and the garlic together in a small bowl or container.
Arrange the potatoes on the baking sheets, evenly spacing them apart. Using a flat tool (like a spatula or the bottom of a glass), smash the potatoes to about ½ inch thick.
Brush the tops of the potatoes with the oil and garlic mixture, then sprinkle them with salt.
Bake the potatoes until the tops are golden brown and the edges are crispy, 35 to 40 minutes.
While the potatoes bake, place the cashews, basil, milk, lemon juice, garlic, and salt into a blender or the bowl of a food processor fitted with an S-blade. Blend until smooth, stopping to scrape down the sides of the bowl as needed.
Taste the mixture and add additional salt or lemon juice, if desired. You can also thin the mixture with additional milk, if desired.
Serve the potatoes immediately after they come out of the oven, with the aioli on the side or drizzled on top.