Combine the soy sauce, fish sauce, sesame oil, brown sugar, red pepper flakes, garlic, scallions and ginger in a large resealable plastic bag. Add the flank steak, seal the bag and massage the marinade into the steak. Marinate in the refrigerator for 2 to 4 hours.
Preheat the oven to 375 degrees F.
For the filling:
Bring a pot of water to a boil over high heat. Add the eggs and boil for 6 minutes. Remove to an ice bath to cool, then peel the eggs. Thinly slice the mushrooms. Mix together the mushrooms, kimchi, panko and scallions in a bowl and set aside.
Remove the steak from the marinade (reserve the marinade) and pat dry. Spread the filling over the steak, leaving a 1 inch border around the edge. Lay the eggs end to end at the edge of the filling closest to you. Pick up the end of the steak with the eggs and roll up to the other side, enclosing the eggs and filling.
Lay half the bacon strips next to one another on a work surface so they overlap slightly. Make a second row with the remaining bacon strips, overlapping the ends to make an extra-long rectangle. Place the steak, seam-side down, perpendicular on one edge of the bacon strips and roll the steak with the bacon. Tie a piece of butcher's twine around each strip of bacon to secure it to the steak, then tie a piece lengthwise around the roll. (This can be done a day ahead, covered and refrigerated.)
Heat a large ovenproof saute pan or skillet over medium-high heat. Sear the rolled steak on all sides until golden brown, about 8 minutes total. Brush with the reserved marinade and roast in the oven until an instant-read thermometer inserted into the center registers 135 degrees F, about 35 minutes, brushing again with the marinade halfway through. Transfer to a cutting board and let rest for 10 minutes.