In a large bowl, beat the eggs with an electric mixer on high until foamy and light yellow, a minute or so. Add the yogurt, oil, sugar, and vanilla and beat again on medium until one even color and consistency, another minute. Set aside. You can also do this with a whisk and a little elbow grease.
Bake until a toothpick inserted into the center comes out clean, 25 to 28 minutes for full-size cupcakes, closer to 20 minutes for minis, and about an hour for a loaf.