Preheat oven to 425°.
Place potatoes in a large bowl with smoked paprika, cumin, 1/2 tsp salt, and 1/4 tsp pepper. Drizzle with olive oil and toss to coat. Transfer to a sheet pan and set bowl aside. Spread potatoes into an even layer and bake for 15 minutes. While baking, add lamb to the empty bowl and crumble in goat cheese. Add a pinch of salt and pepper and gently combine with your hands. Form into two patties that are slightly wider than your burger buns. Set aside.
Flip potatoes and return to oven for another 10-15 minutes, until well browned.
Meanwhile, add butter to a large cast-iron skillet over medium heat. When butter has melted add patties and cook for about 2 minutes per side, or to desired doneness. Remove to a large plate.
Add spring onions to the skillet cut-side-down and cook for 1-2 minutes until browned then flip with tongs and cook for 1-2 minutes more. Remove to the plate with burgers.
Add burger buns to the skillet cut-side-down and toast for 2-3 minutes then remove. Spread a thin layer of Dijon mustard on top and bottom buns. Assemble burgers with bottom bun, burger patty, arugula, spring onions, and top bun. Serve with potato wedges.