Using metal tongs, broil poblano peppers over a gas stovetop until the skins blister, about 3 minutes. Broil all sides of the peppers evenly. Immediately place peppers in a large resalable plastic bag and seal bag. Let sit for 15 minutes.
Peel peppers and remove stems. Throw away the skin, stems, and seeds. Roughly chop peppers.
Heat oil in a large, heavy pot over medium heat. Cook beef until golden brown on all sides. Remove beef with a slotted spoon; set aside.
Cook the onion in the same pot until tender, about 6 minutes. Add beef, poblanos, broth, tomatoes, garlic, ¼ cup cilantro, cumin, chili powder, mustard, salt, and pepper. Increase heat to high to bring mixture to a boil. Reduce heat to medium-low, cover and simmer until beef is tender, about 1 hour and 20 minutes.
Add potatoes and cook for 15 minutes. Skim excess fat from soup. Garnish with remaining chopped cilantro.