In a small pitcher, whisk the milk with the flour until smooth. Melt the butter in a large skillet. Add the bacon and cook over moderate heat, stirring, until crisp, about 3 minutes. Add the onion and garlic and cook, stirring, until softened, about 4 minutes. Whisk the milk again and add it to the skillet. Cook over moderate heat, stirring, until the sauce thickens, about 1 minute. Add the chopped chayote and cook over moderate heat, stirring often, until the excess liquid evaporates, 7 to 8 minutes. Add half of the cheese and the parsley and season with salt and pepper. Season the chayote shells with salt and pepper. Mound the filling in the shells and sprinkle with the remaining cheese. Transfer the stuffed chayotes to a baking sheet.