Preheat oven to 450 degrees. Wash and dry all produce. Dice potatoes into 1/2-inch pieces. Core, deseed, and dice bell pepper into 1/2-inch pieces. Toss potatoes on one side of a baking sheet with a drizzle of oil, salt, pepper, and half the paprika. Toss bell pepper on empty side with a drizzle of oil, salt, and pepper. Roast on top rack until tender, 20-25 minutes. TIP: If needed, divide between 2 sheets; roast potatoes on top rack and bell pepper on middle rack.

Meanwhile, heat a large pan over medium-high heat. Add bacon and cook, turning occasionally, until crispy, 6-10 minutes. (TIP: Adjust heat if bacon begins to brown too fast.) Turn off heat and transfer to a cutting board. Drain all but a thin layer of bacon fat from pan. Once cool enough to handle, roughly chop half the bacon.

While bacon cooks, in a small bowl, combine ketchup and remaining paprika. In a separate small bowl, combine sour cream, a big pinch of salt, and as much hot sauce as you like; stir in water 1 tsp at a time until mixture reaches a drizzling consistency.

Trim and thinly slice scallions, separating whites from greens. In a large bowl, combine beef, scallion whites, salt (we used 3/4 tsp; 11/2 tsp for 4 servings), and pepper. Shape mixture into two patties (four patties for 4), each slightly wider than a burger bun. Heat pan with reserved bacon fat over medium-high heat. Add patties and cook to desired doneness, 3-5 minutes per side. Turn off heat; transfer to a plate. Wipe out pan.

Meanwhile, halve buns. Place cut sides up on a second baking sheet; place gouda on bottom buns. Toast on middle rack until buns are golden and cheese has melted, 3-5 minutes. While buns toast, heat a drizzle of oil in pan used for patties over medium heat. Once hot, crack eggs into pan and cover. Cook until whites are set but yolks are still runny, 3-4 minutes. Season with salt and pepper.

Fill buns with patties, smoky ketchup, hot sauce crema (save a bit of each for the hash), bacon slices, and eggs. Toss together veggies, then divide between plates with burgers. Top veggie hash with chopped bacon and scallion greens, then drizzle with remaining ketchup and crema.