Season beef all over with salt and pepper. Heat 2 tbsp. oil in a large wide pot over medium-high heat. Add beef and cook, turning occasionally, until deep golden brown all over, 12-14 minutes total. Add beef stock, one-fourth of the garlic, and one-third of the white onions, stir well, reduce heat to medium-low, and simmer, covered, until tender, about 3 hours. Remove from heat and set pot aside to let rest, covered, until beef is cool enough to handle. Shred beef with your hands and reserve beef and any remaining cooking liquid in 2 separate bowls; skim off and discard fat from cooking liquid.
Meanwhile, heat 2 tbsp. oil in a medium saucepan over medium heat. Add one-fourth of the remaining white onions and one-third of the remaining garlic and cook until translucent, about 10 minutes. Stir in half of the flour, 1 tsp. of the cumin, and half of the oregano. Cook until fragrant, 3-4 minutes. Add ground chiles and 1 1⁄2 cups water and whisk together. Increase heat to medium-high and bring just to a boil. Whisk well, then reduce heat to medium-low, cover, and cook until smooth and thick enough to spoon, 2-3 minutes. Season to taste with salt and pepper; set chile colorado aside.
Heat 2 tbsp. oil in a small saucepan over medium-high heat. Add one-third of the remaining white onion and half of the remaining garlic, stir well, and cook until translucent, 8-10 minutes. Stir in remaining flour, cumin, and 1⁄4 tsp. black pepper and cook for 2 minutes more. Whisk in chicken stock, scraping up any browned bits from the bottom of the saucepan. Add two-thirds of the poblano peppers and remaining oregano and stir until well combined. Bring just to a boil, then reduce heat to medium-low, cover, and simmer, stirring occasionally, until smooth and thick enough to spoon, about 20 minutes. Transfer contents of saucepan to blender and carefully puree; set chile verde aside.
Heat remaining oil in a large skillet over medium-high heat. Add remaining white onion and remaining garlic and cook until softened, 3-4 minutes. Add shredded beef and cook, stirring occasionally, until well browned, 12-14 minutes. Stir in reserved cooking liquid, remaining poblanos, half of the tomatoes, and lime juice. Reduce heat to medium-low and cook until most of the liquid has evaporated, about 5 minutes more. Season to taste with salt and pepper.
Working with one at a time, heat tortillas in a large skillet, flipping once, until hot, then fill each with one-quarter of the beef and about 1⁄4 cup of the cheese. Roll up burritos snugly, tucking in the ends, then transfer to plates and spoon some of the chile colorado and chile verde over them. Top burritos with remaining cheese, remaining tomatoes, lettuce, and red onions.