Place a large pot of lightly salted water on stove and bring to a boil. Fill a large bowl with ice and cold water and set aside, this will be used to "shock" the asparagus to stop the cooking.
Rinse asparagus and snap off or cut off woody ends. When the water comes to a boil submerge the asparagus for 3 minutes and then immediately remove from boiling water and place into the bowl containing ice water
Heat turkey or chicken broth in pan to just under boiling. Remove from heat and add turkey to broth and cover. Set aside.
In a medium sized saucepan, melt the butter over medium-low heat, and add the flour. Cook stirring for about 5 minutes. Do not to brown. Slowly, and carefully, whisk in the cold milk. Continue whisking until the mixture slightly thickens. Add the nutmeg and cayenne. Continue cooking on medium heat until the sauce simmers. Reduce the heat to medium-low, and simmer, stirring occasionally, for 10 minutes. Turn off the heat, and stir in the cheese, one handful at a time. When all the cheese has melted into the sauce, season with salt and pepper to taste.
Place one slice of bread on each plate. Next remove the turkey from the broth and place on top of the bread. Next place 2 spears of asparagus on top of the turkey. Last, top the sandwich with the cheese sauce.