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Place tongue, onion, bay leaves, cilantro stems, carrot, and garlic in a saucepan just big enough to hold them.
Add chicken broth until mostly covered.
Bring to a boil over high heat, reduce to a simmer, cover with a tight-fitting lid, and cook until completely tender, 4 to 6 hours.
Carefully remove tongue to a cutting board. Strain stock and discard solids, reserving liquid for another use.
Peel the outer membrane off the tongue and discard. Roughly chop tongue into ½-inch pieces.
When ready to serve, heat fat in a large non-stick or cast iron skillet set over medium-high heat until shimmering.
Add tongue pieces and cook, stirring occasionally, until tongue is well-browned on all sides, 5 to 8 minutes total.
To serve, wrap a spoonful or two of tongue in a double layer of corn tortillas. Top as desired.