Add tongue, 1/2 chopped onion, 3 tablespoons salt, 1 jalapeño, and garlic cloves to a large pot filled with water to cover tongue by several inches; bring to a boil over high heat. Reduce heat to medium-low; cover and simmer until very tender, 3 to 4 hours. Remove tongue from pot; allow to cool. Reserve cooking liquid. When tongue cool enough to handle, peel off tough outer skin; shred meat using two forks.

Bring a large saucepan of water to a boil over high heat. Add whole tomatoes and remaining 2 jalapeños; boil until tender. Transfer tomatoes and jalapeños to a blender; purée until smooth.

Heat vegetable oil in a large skillet over medium heat. Stir in remaining 1/2 chopped onion; cook until softened and translucent, about 5 minutes. Stir in shredded tongue and puréed tomato mixture; bring to a simmer, stirring occasionally, then pour in 2 cups reserved cooking liquid. Cook and stir until liquid evaporated, leaving moist, flavorful meat, about 20 minutes.