Preheat the oven to 425 degrees F. Line a rimmed baking sheet with foil. Toss the sweet potato rounds with 1 tablespoon oil and a pinch each of salt and pepper on the prepared pan until well-coated. Spread in a single layer. Roast until browned and tender, about 20 minutes. Meanwhile, form the beef into 12 patties the same diameter as the sweet potato rounds. Sprinkle with salt and pepper. Heat the remaining 1 teaspoon oil in a large skillet over medium-high heat until hot. Add the patties in a single layer. Cook, turning once, until well-browned and cooked through, 4 to 6 minutes. Place 1 patty on each sweet potato round. Top with the cucumber slices, sauce, and arugula.
For Slider Sauces: Stir together all of the ingredients (for each sauce separately) until well-mixed.