Peel and discard the outer, fibrous skin layers of banana heart until you reach the lighter, softer core. Trim off the stem and discard. Cut banana core into half and slice thinly.
In a bowl of cold salted water, place shredded banana heart and soak for about 15 to 20 minutes. Using hands, squeeze to release bitter sap. Rinse with cold water and with hands, squeeze well to dispel liquid.
In a pan over medium heat, heat about 1 tablespoon oil. Add onions and cook until softened.
Add banana heart and cook, stirring occasionally, until cooked through.
Add oyster sauce and stir to combine. Season with salt and pepper to taste.
Continue to simmer until liquid is mostly absorbed.
Remove from pan, drain well, and allow to cool.
In a bowl, combine sauteed banana heart, flour, and eggs. Mix well. Form into 4 uniform-sized patties.
In a pan over medium heat, heat about 2 tablespoons oil. Gently add banana heart mixture in a single layer and cook for about 2 to 3 minutes on each side or until golden and firm. Remove from pan and drain on paper towels.
Assemble burgers with desired condiments, veggie patty, cheese, lettuce, and tomatoes.