Heat 12-inch skillet over medium. Lightly brush tortillas with 1 tablespoon oil; warm individually about 20 seconds per side until flexible. Stack under clean towel.
Raise to medium-high. Add tomatillos cut-side down, onion, unpeeled garlic, and jalapeno to dry skillet. Cook 8 to 10 minutes, turning as needed, until softened and patchily browned. Remove garlic early if skin darkens.
Cool vegetables 5 minutes; peel garlic. Blend with tomatillos, onion, jalapeno, broth, and 1/2 teaspoon salt mostly smooth in vented countertop blender, following warm-liquid instructions. Leave sauce in jar while making filling.
Reduce skillet to medium; add remaining oil, beans, squeezed spinach, cumin, and remaining salt. Cook 3 to 4 minutes, mashing half beans. Transfer to bowl; add half cheese.
Pour half sauce into skillet; lower heat. Divide filling among tortillas; roll and arrange all 8 seam-side down in one snug layer. Spoon remaining sauce over.
Cover; gently simmer 10 to 12 minutes to 165°F in centers. Add remaining cheese; cover 2 minutes until melted. Remove from heat.
Drizzle lime juice around rolls; rest 5 minutes. Serve 2 with sauce per portion.