For the buns: Stir the water, yeast and sugar together to activate the yeast. (You can tell when the yeast is activated-little bubbles will have formed, which will take a minute or two.) Add the melted butter and 3 of the eggs and mix until incorporated.
Add 4 cups flour and the salt to the bowl of a stand mixer fitted with the dough hook attachment. Mix together on low speed, then stop and add the wet ingredients. Continue to mix until the dough is very smooth, about 5 minutes. You may want to add a little bit more flour as you mix; the dough should be sticky but workable.
Transfer the dough to a bowl. Cover the bowl with a kitchen towel, place in warm area and let rise until the dough has doubled in size, about 30 minutes.
After the dough has proofed (risen), turn it out onto a lightly-floured countertop and knead in the Cheddar and black pepper. Try not to overwork the dough; once the cheese is mixed in you can stop. Cut the dough into six equal portions and shape them into round balls. Place on a buttered baking sheet and flatten them with your hand.
Cover and let rise again in a warm area until they have doubled in size, about 30 minutes more. When they are almost doubled in size, preheat the oven to 350 degrees F.
Beat the remaining egg in a small bowl and brush the tops of the rolls with the egg. Bake until golden brown, rotating halfway through, 10 to 15 minutes. Cool on a wire rack.
For the ketchup: Turn the oven down to 250 degrees F. Cut the garlic in half crosswise and sprinkle with olive oil, salt and pepper. Wrap in foil and roast until soft, 35 to 40 minutes. Remove the garlic from the skins and mix into a paste. Reserve 1 tablespoon for the ketchup, and use remaining garlic for another use.
Whisk together the mayonnaise, mustard and roasted garlic in a bowl. Add the ketchup and vodka and mix until incorporated. This will keep for up to 6 days in the refrigerator.
For cooking the burgers: Heat a skillet over high heat or heat an outdoor grill to high. Sprinkle the patties with salt and pepper. Coat with the oil and sear or grill until medium rare, 3 to 5 minutes per side.
Slice the buns in half horizontally and brush with the melted butter. Toast in the skillet or on the grill. Arrange 8 pickle slices on each bottom bun and place a hamburger on top. Slather 1 to 2 tablespoons of spiked ketchup over the burger and set the top bun in place. Serve immediately with the remaining spiked ketchup on the side.
This recipe makes very large buns and a large quantity of ketchup. Use half the dough to make extra buns for another use, or halve the dough recipe. Save extra ketchup for other recipes, or halve the amounts for less.