For the mustard: Combine the mustard, sugar, vinegar, olive oil, horseradish, lemon juice, salt and 1/2 cup water in a blender
or food processor
and process for 1 minute. Scrape down the sides of the container and process for 30 seconds. Transfer to a small saucepan and let stand for 10 minutes.
Set the saucepan over low heat and cook until bubbly, stirring constantly. If a thinner mustard is desired, stir in an additional 1 to 2 tablespoons water. Let cool completely.
For the wienies: Grill or boil the hot dogs. Cut off the tips, and then cut the hot dogs into 1-inch pieces. Place a toothpick into each piece.
Spoon a thin layer of mustard into each shot glass. Place the hot dogs into the mustard in the shot glasses.