Heat the chile pepper and garlic in a dry 10-inch cast-iron skillet over medium-high heat, turning occasionally, until the chile is pliable and the garlic is browned in spots, about 5 minutes. Remove from the heat and let cool slightly; remove the stem from the chile and peel the garlic. Transfer the chile and garlic to a blender (reserve the skillet). Add the tomato sauce, chicken broth and cumin; puree until almost smooth.
Heat 1 tablespoon vegetable oil in the reserved skillet over medium heat. Slowly add the tomato mixture (it will splatter) and cook, stirring occasionally, until slightly thickened and darkened, 8 to 10 minutes. Season with salt and black pepper.
Meanwhile, heat the remaining 2 teaspoons vegetable oil in a medium nonstick skillet over medium heat. Crack the eggs into the skillet and season with salt and black pepper. Cook, without flipping, until the whites are set but the yolks are still runny, 3 to 4 minutes.
Gently stir the tortilla chips into the tomato sauce in two batches until coated. Remove from the heat and top with the eggs, radishes, red onion, cilantro and/or Cotija cheese.