Preheat your oven to 350°F. Grease or line a standard 9-inch x 5-inch loaf pan and set it aside.
In a medium-sized bowl, place the flour, xanthan gum, baking powder and salt, and whisk to combine well.
Add the grated zucchini, and the Parmigiano-Reggiano and cheddar cheeses, and mix to combine, breaking up any clumps in the grated zucchini or cheeses.
In a blender or medium-size bowl, place the oil, eggs and yogurt, and blend (or whisk vigorously) until smooth.
Create a well in the center of the dry ingredients, and pour in the wet ingredients. Mix until well-combined.
The batter will be thick. Scrape it into the prepared pan and, with a wet spatula, smooth into an even layer. With a sharp knife, slice about 1/4-inch deep down the center of the loaf.
Place the pan in the center of the preheated oven and bake until lightly golden brown on top and a toothpick inserted in the center comes out clean (about 40 minutes).
Allow to cool for at least 10 minutes in the pan before turning out onto a wire rack to cool completely. Slice and serve.
Once completely cool, the sliced (or unsliced) bread can be wrapped tightly and frozen for longer storage.