Carefully place the 6 large eggs in a single layer at the bottom of a saucepan. Cover the eggs with at least 1 inch of cold water. Bring the water to a rolling boil over high heat, then immediately remove the pan from the heat, cover it, and let it stand for 8 minutes. After 8 minutes, transfer the eggs to an ice bath for 5 minutes to halt the cooking process and make them easier to peel. Once cooled, gently peel each hard-boiled egg under cool running water.
Using a sharp knife, carefully slice each peeled egg lengthwise in half. Gently scoop out the bright yellow yolks into a medium bowl, placing the pristine white halves onto a clean serving platter, cut-side up.
Add the mayonnaise, Dijon mustard, white vinegar, salt, and freshly ground black pepper to the bowl with the egg yolks. Mash the mixture thoroughly with a fork until it is completely smooth and free of lumps. The texture should be creamy and uniform.
If desired, add 2 drops of orange gel food coloring to the yolk mixture. Stir vigorously until the color is evenly distributed and the mixture resembles a vibrant pumpkin orange hue. Taste the filling and adjust seasonings as needed, adding more salt or pepper to your preference.
Spoon the orange yolk mixture into a piping bag fitted with a star tip (or use a zip-top bag with a corner snipped off for a simpler presentation). Carefully pipe or spoon an even amount of the filling into each egg white cavity, mounding it slightly to create a plump, rounded "pumpkin" shape.
Gently dust each filled egg with a light sprinkle of smoked paprika. For an authentic pumpkin look, use the tip of a small knife or a toothpick to create several shallow vertical lines on the paprika-dusted filling, mimicking the natural ridges of a pumpkin.
Take the small, 1-inch pieces of fresh chives and gently insert one piece vertically into the top center of each deviled egg, serving as the "stem" for your pumpkin. Cover the platter loosely with plastic wrap and refrigerate the Pumpkin Patch Deviled Eggs for at least 30 minutes to allow the flavors to meld and the eggs to chill thoroughly. Serve cold.