Preheat oven to 350˚F (180˚C). Line 2 baking sheets with parchment or lightly grease.
In the bowl of a large food processor fitted with a sharp blade, combine the flour, baking powder, and salt.
Add the butter and pulse to create a coarse texture with no pieces larger than a pea.
Add the finely grated carrots, cheddar, parmesan, and egg yolk. Continue to process until the dough comes together.
On a lightly floured surface, roll the dough into a thin sheet about 1/8 inch (3 millimeters) thick.
Cut the sheet into squares about 1 inch (2.5 centimeters) thick. Prick the center of each square with a fork.
Arrange the squares 1/2 inch (1.25 centimeters) apart on the prepared baking sheets.
Bake in the preheated oven until golden on the bottom, about 13-15 minutes. Allow to cool before removing to plate. Repeat with remaining dough.
Allow to cool before removing to a plate.
These Carrot Cheese Crackers are best the day they are made. Store leftovers at room temperature in an airtight container.