In a large bowl, combine flour, milk, shallots, parsley, baking powder, salt, black pepper, and thyme. Gently fold in the carrots; be careful not to overmix. Cover, and refrigerate for 3 hours.
Heat vegetable oil in a deep fat fryer to 375 degrees F (190 degrees C). Lower the batter by heaping tablespoonful into the hot oil. Do not crowd the fritters, fry only a few at a time. Fry until fritters are golden brown, about 10 to 12 minutes. Remove fritters with a slotted spoon and drain well on paper towels. Serve immediately with Bearnaise sauce.