Remove the stems from the chiles de árbol. Remove the stem, ribs, and seeds from the poblano; finely chop the pepper. Wash your hands after handling. In a medium pot, combine the chiles de árbol, tomatoes, garlic, and half the poblano with the oil, 1 cup water, and ½ teaspoon salt (reserve the remaining poblano for garnish). Bring to a boil, reduce to a simmer, and cook until the chiles are soft, 8 to 10 minutes.

Using a slotted spoon, transfer the chiles de árbol, tomatoes, and garlic clove to a blender. Let cool to warm, then puree. Return the puree to the pot.

Bring the sauce to a simmer. Crack the eggs directly into the sauce, cover, and cook over low heat, basting the eggs with the sauce a few times, until the whites are set but the yolks are still runny, 4 to 5 minutes. While the eggs cook, prepare the garnishes.

Using a spoon, scoop the avocado flesh out of its peel; thinly slice the avocado. Coarsely chop the cilantro.

Using a slotted spoon, transfer the eggs to individual plates. Sprinkle lightly with salt and drizzle with olive oil. Ladle the sauce on top. Garnish with the avocado slices, cilantro, and remaining chopped poblano and serve with tortillas or crusty bread.