Preheat your broiler to high. Spread the tomatillos, onions, garlic, and green onions on a foil-lined sheet pan. Broil, tossing occasionally, until the vegetables are softened and blackened (around 8–10 minutes depending on heat level).

Transfer roasted vegetables to a blender. Add oregano, cilantro, Hatch chile powder, salt, and pepper. Add a splash of water if needed. With the blender running, slowly drizzle in the olive oil to emulsify. Puree until smooth. Set aside.

Stack the corn tortillas, slice into 3 even strips, then cut each into rectangles (~60 pieces).

Heat neutral frying oil in your skillet over medium-high heat. Test with one piece—if it bubbles immediately, you're ready. Fry tortillas in batches until golden and crispy, about 1–2 minutes each batch. Transfer to paper towels to drain.

Wipe the skillet clean (if needed) and pour in the green chile sauce. Bring to a simmer over medium heat. Add tortilla pieces, stirring gently to coat. Cook just enough to soften the chips—about 2–3 minutes—so they still hold some crunch.

Make 4 wells in the chile‑coated chips. Crack an egg into each and gently fry or poach until whites are set but yolks still soft. Remove pan from heat.

Sprinkle crumbled cheese across the dish and let it melt slightly. Dollop Mexican crema over each serving.

Top with cilantro leaves and shaved radish slices for color and crunch. Serve immediately while hot and saucy.