Make the sauce first. Cook all the vegetables except the beans in the olive oil until they're tender. Add the beans, salt, and enough water to cover by about an inch. Simmer until the beans are tender--about two hours.
When the beans are tender enough to squash with thumb and forefinger, add the sherry. Then puree the mixture in a blender. Add a little water if necessary to give the beans a sauce consistency. Keep warm.
In a heavy skillet over medium heat, heat the butter until bubbling and add shallots, garlic, mirliton, squash, and Creole seasoning. Cook until the vegetables soften. Cool to lukewarm, then add the cheese.
Place one-sixth of the stuffing atop a pounded chicken breast. Roll it up and secure with a toothpick.
Combine the salt and pepper with the flour. Dust the chicken rolls with the seasoned flour.
Heat about a quarter-inch of oil in a heavy skillet until it shimmers. Fry the chicken rolls, turning every thirty seconds or so, until browned and the inside is hot.
Allow the rolls to cool for about five minutes. Slice the rolls on an angle into three or four slices.
Spoon about 1/4 cup of the black bean sauce onto warmed plates. Top with the chicken rolls. Garnish with the green onions, cilantro, and crumbled cheese.