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.
This recipe has been updated to more accurately recognize its origin or to add cultural context. It may differ from what was originally published or broadcast.
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