In medium saute pan, add 2 ounces olive oil, onions, garlic, and peppers and saute until translucent. Add chicken and brown on all sides. Remove half of the onion and pepper mixture.
Continue to cook chicken and deglaze with beer, water, and lime juice, salt and pepper, let simmer for 45 minutes until chicken begins to fall off the bone. Remove chicken, let cool. Remove bone and remaining skin, and shred with forks.
In same saute pan, reduce mixture until thick. Remove mixture from the pan, add the remaining 2 ounces oil and potatoes and fry potatoes until crispy. Add onion and pepper mixture and shredded chicken. Press mixture together as it cooks and form into 1/2-inch cake. Flip over to brown both sides.