Heat a 12-inch skillet over medium heat. Brush tortillas lightly with 1 tablespoon oil and warm each for about 20 seconds per side until flexible, then stack under a clean towel.
Raise the heat to medium-high. Add tomatillos cut-side down, onion quarters, unpeeled garlic and jalapeno to the dry skillet. Cook for 8 to 10 minutes, turning as needed, until vegetables are softened and browned in patches, removing garlic earlier if its skin darkens.
Let charred vegetables cool for 5 minutes, then peel the garlic. Blend garlic, tomatillos, onion, jalapeno, broth and 1/2 teaspoon salt in a vented countertop blender until mostly smooth, following its warm-liquid instructions. Keep the sauce in the blender jar while preparing the filling.
Lower the skillet heat to medium and add remaining oil, black beans, squeezed spinach, cumin and remaining salt. Cook for 3 to 4 minutes, mashing about half the beans, then transfer to a bowl and stir in half the cheese.
Pour half the tomatillo sauce into the skillet and lower the heat. Divide filling among tortillas, roll and arrange all 8 seam-side down in one snug layer, then spoon remaining sauce over the rolls.
Cover and simmer gently for 10 to 12 minutes until the centers reach 165°F. Scatter remaining cheese over the top, cover for 2 minutes until melted and remove from the heat.
Drizzle lime juice around the rolls and rest for 5 minutes. Serve 2 enchiladas with sauce per portion.
