To make the sliders place the olive oil and garlic in a sauté pan set over medium-high heat. When the garlic begins to sizzle (about 45 seconds), add the onion and sauté until the onion becomes limp and translucent, about 3 minutes. Add both paprikas, salt and cumin and stir to combine.
Place the onion and garlic mixture, ketchup, and Worcestershire sauce in a blender and puree until smooth. Place the beef and pork in a medium bowl and pour the puree over the meat. Using your hands, work the puree into the meat until well mixed. Allow the meat mixture to rest in the refrigerator for about 1 hour.
Roll about 1/4 cup of the meat mixture into a ball and lightly flatten into a patty about 1/2-inch thick. Repeat with the remaining mixture.
Heat a griddle or sauté pan over medium-high heat. Lightly coat the pan with vegetable oil. When the oil is hot, cook the patties for about 3 minutes per side.
To make the potato sticks, line a dish with paper towels and set aside. Pour 2 inches of oil into a deep, heavy-bottomed pot. Place the pot over high heat and wait for the oil to reach 375°F on a deep-fry thermometer. If you don’t have a deep-fry thermometer, you can test the oil temperature by dropping a couple of potato sticks in the oil—if bubbles rise up around the potatoes, it’s ready.
Sprinkle a small amount (about a four-finger pinch) of the potato sticks into the hot oil Small batches keep the potatoes from forming into one large mass.
When the potatoes cook to a golden brown color, in about 2 minutes, remove them from the oil with a slotted spoon and place on the paper towel–lined dish to drain. Immediately season with salt. Repeat with the remaining potato sticks until all are cooked, drained, and salted. Some of the potato sticks will have stuck together. When they have cooled, gently break apart.
To assemble the sliders layer a mound of potato sticks on the bottom bun, place a patty over it, then top with another mound of potato sticks and the other half of the bun. Serve.