Preheat oven to 425 degrees and rub the potatoes generously with olive oil. Sprinkle the exterior of each potato generously with kosher salt then place on a baking sheet. Poke each of the potatoes with a fork or paring knife and bake 50–60 minutes or until fork-tender. Remove potatoes from oven and set them aside until cool enough to handle.

Once potatoes are cool enough to touch, slice each one in half horizontally. Beginning with the tops, remove the flesh from the skin and add the flesh to a medium-sized mixing bowl and discard the skins. For the bottoms of the potato, scoop out the very center portion of the flesh but make certain to leave at least a ¼" edge in all directions to support the filling. Add the scooped-out flesh to the bowl.

Mash the potatoes in the bowl with a fork until no large chunks remain then add the yogurt, olive oil, dried basil and cheese and mix until smooth. (You can add a little more olive oil or yogurt if you find the filling to be a little too dry for your taste. Potatoes vary greatly in size, so you’ll have to adjust to your liking.)

Fold in the chopped tomato and season to taste with salt and pepper. Divide the filling between each of the potato boats, sprinkle with a little bit of extra cheese and return potatoes to oven. Bake an additional 15 minutes or just until the cheese and filling are nice and hot.