Preheat the oven to 225 degrees F.
Place the brioche rolls and their tops on a baking sheet and place in the oven to warm.
In a large skillet, heat the oil over medium-high heat. Add the ham and cook, stirring, until lightly caramelized, about 3 minutes. Add the onions, carrots, and celery and cook, stirring often for 4 minutes. Add the mushrooms and cook for 5 minutes. Add the garlic, salt, and pepper, and cook for 30 seconds. Add the chicken and cook, stirring often, until slightly caramelized, about 5 minutes.
Add the stock and bring to a boil. Cook until reduced by half, about 3 minutes. Add the cream and cook until reduced by half.
Add the butter to the hash and stir to incorporate. Add the thyme and Brabant potatoes, and toss to combine. Remove from the heat.
Have a pot of simmering water. Add vinegar. Carefully break the eggs into a small bowl. Slide the eggs, 1 at a time, into the water and poach just until the white is set and the center is still soft, about 3 minutes. Remove the poached eggs from the water with a slotted spoon and place on a paper towel lined plate.
Remove the brioches from the oven and place 1 on each of 4 plates. Spoon the chicken hash into each brioche, sprinkle with cheese and top with poached eggs. Set the brioche tops on each at an angle, and sprinkle with the parsley. Serve.
Yield: 4 servings