In a medium size, nonreactive saucepan, combine the water, salt, and vinegar. Bring to a boil, add the carrots, and simmer until just tender, 5 min. Remove them from the water, shock under cold running water, and reserve. Repeat with the potatoes, cooking them about 3 min. Heat 1 Tbs. of the oil in a medium skillet over medium-high heat. Add the carrots, potatoes, and sausage. Sauté briskly until the sausage is crisp and the vegetables are browned, 10 to 12 min. Set aside.
Heat the oven to 400°F. If the sauce has been chilled, heat it until just warm and thin it with a bit of water if necessary. Heat a skillet over medium high heat. With a pastry brush, lightly coat both sides of a tortilla with the sauce. Pour about 1 Tbs. oil into the pan, swirling to coat. The pan should be lightly and evenly filmed with oil. Set the tortilla in the pan and cook until it just begins to brown, 10 to 20 seconds, pressing down with a spatula when the tortilla puffs. Flip the tortilla over and cook the other side in the same way. When cooked on both sides, each tortilla should be very pliable but not soft enough to tear when handled. The tortilla will begin to crisp around the edges, but don’t let it become so hard that it can’t be easily rolled. Drain on paper towels. Add more oil to the pan if needed; repeat with the remaining tortillas.
Brush a thin layer of sauce on the tortilla-just enough to coat.
Use a spatula to flip the softened tortilla. Tongs would tear it.
Mix 3/4 cup of the grated Jack or Cheddar with the feta. Arrange a heaping Tbs. of the cheese mixture just off center of a tortilla and loosely roll or fold the tortilla into a cylinder. Repeat with the remaining cheese and tortillas, setting the rolled enchiladas side by side and seam side down in a 9x13 inch baking dish as you go. Spoon a scant cup of sauce over all, sprinkle with the remaining grated cheese, and then sprinkle on the carrots, potatoes, and sausage. Bake for about 7 min. and then drizzle on about 1/2 cup more sauce. Continue baking until the cheese is melted and starts to bubble, about 10 min. Serve.
Roll enchiladas loosely and don't overfill them. A heaping tablespoon of filling is all you need.