Preheat the broiler with an oven rack in the middle of the oven. Place tomatoes, cut sides up, onion, and garlic, on a rimmed baking sheet, and broil in preheated oven on the middle rack until charred, about 15 minutes. Let cool.
Bring water to a boil in a small saucepan over medium-high. Add dried chiles, and boil, stirring occasionally, until chiles are softened, about 10 minutes. Drain, reserving 1 cup of liquid.
Combine tomatoes, onion, garlic, chiles, oregano, salt, reserved cooking liquid, and cilantro stems in a blender. Process until smooth, about 20 seconds. Pour mixture through a fine wire-mesh strainer into a bowl; discard solids. Add red chile sauce to skillet over medium. Bring to a simmer.
Heat butter in a nonstick skillet over medium. Crack eggs into skillet. Cook until edges are set and starting to brown, about 2 minutes. Cover skillet, and cook until whites are set.
Pour red chile sauce over chips in a large bowl; toss to coat. Divide the chip mixture evenly among plates. Top evenly with eggs, avocado slices, cotija, crema, and cilantro leaves. Serve immediately with apple juice.