Preheat oven to 425°F. Combine the shredded zucchini and carrots in a bowl and toss with ¼ cup of flour to coat. Set aside.
In a large bowl, combine the remaining flour, baking powder, salt, and pepper and stir with a fork to combine. Grate the butter and add to the dry ingredients. Stir with a fork until crumbs form.
Add ⅔ cup milk and stir again. Gradually stir in additional milk, just a tablespoon or so at a time. Add just enough milk for the dough to come together and become a tiny bit sticky.
Add the shredded vegetables, parsley, and green onions. Stir to combine. Add the cheese and stir once more to mix throughout the dough. Scoop the dough onto a baking sheet in approximately ¼-⅓ cup portions.
Bake for 14-16 minutes until lightly browned on top and golden brown on the bottom.
Notes
Please note that these biscuits are not baked on parchment. The contact with the metal pan is what helps create that fantastic golden bottom crust for the biscuits.