Combine the eggs and 1 teaspoon salt in a medium bowl and whisk to combine; set aside. Heat 2 tablespoons oil in a large nonstick skillet over medium-high heat until shimmering. Sprinkle the potatoes with salt and add to the skillet in a single even layer. Cook until golden and crispy on the bottom, about 5 minutes. Flip with a heatproof spatula and drizzle 2 more tablespoons oil around the edge of the skillet, giving the skillet a good shake once all the oil is in. (This allows you to use less oil while still getting a crispy potato cake.) Continue cooking until golden and crispy on the bottom, about 5 minutes.
Meanwhile, spread 4 tortillas out on a clean work surface and put 1/2 cup cheese in the middle of each. Using the spatula, break the potato cake into 4 pieces and place on top of the cheese. Wipe the skillet out, return to the heat and add 1 tablespoon oil, then the onion and poblanos. Sprinkle with 1 teaspoon salt and cook, stirring occasionally, until wilted and golden, 8 to 10 minutes. Stir in the spinach and cook until heated through, about 2 minutes. Stir in the eggs and cook, stirring occasionally, until the eggs are just set but still moist, 3 to 4 minutes. Divide the egg mixture evenly among the tortillas and wipe out the skillet. Return the skillet to the heat and reduce it to medium-low.
Fold the sides of each tortilla in, then roll each one up into a burrito. Brush the burritos all over with the remaining tablespoon oil and put them in the skillet seam-side down. Cook until golden and crispy, about 4 minutes, then flip and repeat on the opposite side. Serve with sour cream and hot sauce on the side.
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