In a large saucepan, bring 4 quarts of salted water to a boil. Add the chayotes, cover, and cook until they are tender when pierced with a fork, about 30 minutes.
While the chayotes are cooking, prepare the sauce. In a small bowl, whisk together the flour and 1/2-cup of skim milk until smooth. In a large non-aluminum saucepan, stir together the remaining skim milk and the evaporated skim milk. Bring to a simmer over medium heat. Very gradually whisk in the flour mixture. Stir until the sauce boils and thickens. Reduce the heat to very low and cook for 2 minutes, stirring occasionally. Stir in the sour cream, parsley, and scallions and season to taste with salt (optional) and plenty of pepper.
Drain the cayotes and cut into 1/4-inch thick slices. Put the sliced chayotes in a serving bowl, pour the sauce over, and serve immediately.