Place the potatoes in a large saucepan and cover them with water by 1 inch. Season generously with salt, cover with a tightfitting lid, and bring to a boil over high heat.
Remove the lid and cook the potatoes until a sharp knife easily pierces them and they easily fall off the knife, about 10 to 15 minutes.Using a slotted spoon, transfer the potatoes to a baking sheet. Use the bottom of a measuring cup to smash each potato until it bursts and is flattened to about 1/2 inch thick; set the potatoes aside.
Heat the oil in a large frying pan over medium-high heat until shimmering. Meanwhile, line a small plate with paper towels and set it aside.
Add the shallots to the pan and fry, stirring often, until browned, about 3 to 5 minutes (be careful not to burn the shallots, as some pieces may brown faster than others).
Remove with the slotted spoon to the prepared plate, season with salt, and set aside.
Add the curry powder to the pan and stir until fragrant, about 20 seconds. Using your hands, carefully add the smashed potatoes to the pan. Use a metal spatula to flatten them into an even layer. Season with the measured salt and cook undisturbed until browned and crusty on the bottom, about 6 to 8 minutes. Flip, press into an even layer, and cook undisturbed until browned on the other side, about 4 minutes more.
Sprinkle the fried shallots over the potatoes and toss to combine. Taste and season with additional salt as needed.
Serve immediately, passing the yogurt on the side.
