Heat the oven to 160°C (325°F). Toss the pork with the salt, cumin, oregano, and pepper. Place in a heavy ovenproof pot with the onion, garlic, orange juice, lime juice, and bay leaf.
Cover tightly and braise for 2 hours 30 minutes, until the meat shreds with almost no pressure and a fork twists in easily. Remove the lid and let it cool for 15 minutes.
Lift the pork onto a rimmed baking sheet, discarding the onion, garlic, and bay leaf. Shred with two forks and spread in an even layer. Spoon over 3 to 4 tablespoons of the fat and juices from the pot.
Heat the broiler to high. Broil 6 to 8 minutes, stirring once, until the edges are deep brown and crackling and the pork smells toasty.
While the pork broils, toss the potatoes with the oil and a pinch of salt. Cook in a large skillet over medium-high heat, stirring only every 2 minutes, for 12 to 15 minutes, until the cubes are golden and crisp outside, soft inside.
Whisk the eggs with a pinch of salt. Melt the butter in a nonstick skillet over medium-low heat. Pour in the eggs and stir slowly for 2 to 3 minutes, until they form soft, creamy curds that are barely set. Remove from the heat before they look done, since they continue to cook inside the burrito.
Warm the tortillas in a dry skillet for 15 seconds per side, until soft and pliable.
Layer each tortilla with cheese, eggs, potatoes, and about 1/2 cup (about 100g) carnitas. Fold in the sides, roll tightly, and toast seam-side down in a dry skillet over medium heat for 2 minutes per side, until golden and crisp. Serve with salsa and avocado.