Poke holes in the potatoes, all around. Wrap in a single, damp paper towel. Microwave for 5 minutes on high. Flip over and microwave for 5 minutes more or until tender. Do the same for the second potato. While they are still hot, cut the potatoes in half and scoop out the middle into a 1-cup measuring cup. You need 1 cup of packed, cooked potato. If you need more, microwave the third potato to make up the difference. Run the 1 cup of potato through a potato ricer and smash with a fork.
In a bowl, whisk together the eggs. Add the potato and stir vigorously. Stir in the corn, yogurt, salt, and pepper and add 2 tablespoons of all-purpose flour until a thick batter forms. If the batter is runny, add the third tablespoon of flour.
Spray a nonstick pan with cooking spray and heat over medium heat. Drop 2 teaspoons’ worth of batter in the pan for each fritter, pressing down to make a round of 1½ to 2 inches. Fry until golden brown and then flip the fritters and continue frying until golden brown on the other side, about 1 to 2 minutes on each side.
Toss the tomatoes, balsamic vinegar, and olive oil together in a small bowl. Season with salt and pepper to taste.
Crumble or smear about a teaspoon of goat cheese on a fritter and then top with tomato along with a basil leaf.