Preheat your broiler to high. Spread the tomatillos, onions, garlic, and green onions on a foil-lined sheet pan. Broil for about 8-10 minutes, tossing occasionally, until softened and beautifully blackened.

Blend the roasted vegetables in a blender. Add oregano, cilantro, Hatch chile powder, salt, and pepper. With the blender running, drizzle in the olive oil to emulsify. Puree until silky smooth, then set aside.

Prepare your tortillas by stacking them, slicing into 3 even strips, and then cutting each strip into rectangles (~60 pieces).

Heat neutral frying oil in your skillet over medium-high heat. Fry the tortilla pieces in batches for about 1-2 minutes until golden and crispy. Transfer to paper towels to drain excess oil.

Simmer the green chile sauce in the skillet, bringing it to a gentle simmer over medium heat. Add the crispy tortilla chips, stirring carefully to coat them with the sauce. Cook just until softened, about 2-3 minutes.

Make 4 little wells in the saucy chips. Crack an egg into each well and gently fry or poach until the whites are set but the yolks remain soft.

Sprinkle crumbled cheese across the chilaquiles, allowing it to melt slightly. Add a dollop of Mexican crema over each serving.

Garnish with fresh cilantro leaves and shaved radish slices. Serve immediately and enjoy while hot.