Preheat the oven to 450°F and grease a large rimmed baking sheet with cooking spray or a thin layer of oil.
Place the cleaned small red potatoes in a medium pot and cover with cold water; add 1 tablespoon salt. Bring to a simmer and cook until just fork-tender, about 10 to 15 minutes.
Drain the potatoes in a colander and let them cool for about 5 minutes so they are easy to handle.
Drizzle 1 tablespoon of the olive oil over the prepared baking sheet. Place the potatoes on the sheet and toss or shake so they are lightly coated in oil.
Use a potato masher or the bottom of a flat-bottomed glass (sprayed with cooking spray) to gently smash each potato to about 1/2-inch thickness.
Drizzle the remaining 2 tablespoons olive oil over the smashed potatoes, season with additional salt and pepper to taste, and spread them in a single layer with some space between pieces.
Bake until the potatoes are golden and crispy around the edges, about 30 to 40 minutes.
While the potatoes bake, make the aioli by stirring together the yogurt or mayonnaise, pressed garlic, lemon juice, 1/8 teaspoon salt, and 1/8 teaspoon pepper in a small bowl.
Serve the crispy smashed potatoes warm with the aioli for dipping.