Combine the oregano, cumin, salt, pepper, and allspice in a small bowl.
Coat the chuck roast on all sides with the spices.
In a Dutch oven over medium heat, toast the chiles until fragrant, about 2 minutes. Remove from the pot.
Add the oil to the pot.
Add the beef and sear on all sides, about 4 minutes per side.
Once seared, remove the pot from the heat and add the garlic, vinegar, beef broth, and toasted chiles.
Cover and cook for 4 hours, or until pull-apart tender. Halfway through cooking, check the pot and add beef broth as needed.
Shred the beef using two forks.
In a deep skillet or large pot, heat the vegetable oil over medium heat until a thermometer reads 350 F.
While the oil is heating, roll the flautas. Working in batches of 5, wrap the tortillas in damp paper towels and microwave to soften, about 20 seconds.
Add ¼ cup of beef to a tortilla.
Tightly roll the tortilla into a cylinder and secure with a toothpick. Repeat with the remaining beef and tortillas.
Add the flautas to the oil in batches of 4 or 5 at a time and fry until golden brown, turning as needed, for about 4-5 minutes.
Remove the flautas and drain them on paper towels.
To serve, top with sour cream, cilantro, and sliced radishes.