In a large pot, place the meat, the half of the onion you haven't chopped, garlic cloves, bay leaves, and 1 teaspoon salt. Cover with water, filling up to 2 inches above the meat, and bring to a rolling boil over high heat. Skim off any foam that rises to the surface, then reduce heat to medium; cover, and simmer for 1 to 1 hour 15 minutes, until the meat is thoroughly cooked, soft, and can easily shred.
Strain the meat, discard the liquid, and set aside. Once the meat has cooled, shred it into fine pieces using two forks or a stand mixer.
Heat the oil in a large skillet set over medium heat. Add the remaining chopped onion and the jalapeños, stirring occasionally, for 3 to 4 minutes, until they start to soften. Add the tomatoes and 1 teaspoon salt, and cook for 8 to 10 minutes. Stir in the shredded meat and cook for a couple more minutes so it all comes together.
Meanwhile, in a medium mixing bowl, crack the eggs, season with the remaining ½ teaspoon salt and black pepper, and whisk until foamy.
In the same skillet with the meat and veggies, pour in the whisked eggs and cook over medium-low heat. Remove from heat.
On a comal, griddle, or nonstick skillet, heat the tortillas over medium heat. To assemble the burritos, add a generous amount of filling in a thick strip to one side of the tortilla, tuck in the short sides over the filling, then fold the bottom of the tortilla and roll away from you. Repeat with the remaining tortillas.
Serve immediately or wrap each burrito in foil and take them to-go.