Combine the chuck roast, enchilada sauce, adobo sauce, beef broth, quartered onion, garlic, cumin, oregano and bay leaves in a large pot.
Bring to a simmer over medium heat, then cover, reduce heat to low and cook 1 hour 30 minutes, until the meat shreds easily when pressed with a fork.
Lift the meat out, shred it with two forks, and strain the broth, reserving it as the dipping consomme.
Heat the oven to 425°F and line a large sheet pan with parchment paper.
Soak the cut potatoes in cold water 15 minutes, drain, and pat completely dry.
Toss the fries with oil, cornstarch and salt, spread in a single layer, and bake 35 to 40 minutes, flipping once, until deep golden and crisp at the edges.
Return 1.5 cups of the shredded birria meat to a skillet with 1/2 cup of the reserved consomme and warm over medium heat 5 minutes, until glossy.
Pile the hot fries onto a platter and scatter the mozzarella over them while hot, letting it melt into the gaps.
Top with the warmed birria meat, cilantro and diced onion, and serve with lime wedges and a small cup of consomme for dipping.
Good to Know
Braise the meat up to 2 days ahead and store it in its broth; the flavor deepens overnight and day-of prep drops to just the fries.