Preheat oven to 325°F.
Grease (very generously) 8 1/2 x 4 1/2-inch loaf pan with cake release, or grease with shortening and then flour. Set aside.
Grate carrots on a fine grater. Set aside.
Add flour, sugar, cinnamon, baking soda and salt to a large mixing bowl.
Whisk the dry ingredients together until combined.
In a separate medium mixing bowl, add vegetable oil, melted butter, Greek yogurt, eggs and vanilla extract.
Whisk ingredients together until combined.
Add wet ingredients to the dry ingredients in the large mixing bowl.
Mix ingredients together until just combined and no dry ingredients remain. Do not overmix.
Fold carrots into batter.
Add batter to prepared loaf pan and bake for 50-65 minutes until a toothpick inserted into the center comes out with just a few crumbs attached. (If your cake is browning too fast, you can use a foil tent. Just fold a piece of foil in half and loosely place over the top of the cake.)
Remove cake and let cool on a cooling grid for 10 minutes in the pan.
Loosen cake from sides of pan with a small spatula and turn out onto the cooling grid. Cool cake for 1 hour or more until completely cool.