In a large Dutch oven, bring 4 inches of water to a simmer over medium heat. While the water is coming to a simmer, make the crostini. Cut out rounds from each slice of bread using a 3-inch round cookie cutter. Lightly toast the 8 rounds in a toaster and spread the goat cheese evenly on each. Once the water is simmering, crack and slide each of the 8 eggs into the water. Adjust the heat to keep the water at a simmer. Poach the eggs until the whites are completely set and the yolks still feel soft to the touch. Remove the eggs from the water with a slotted spoon and place on a paper towel to drain.
To assemble, place 2 crostini on a plate and place a poached egg on each. Spoon 3 generous tablespoons of the Stewed Bell Peppers evenly over the tops of both eggs. Garnish with a sprig of thyme.
Cut each bell pepper in half through its stem and remove the core, seeds, and ribs. Thinly slice each pepper half into 1/8-inch thick slices.
In a medium saucepan, heat the olive oil over medium-high heat. When the oil is hot, add the slivered garlic and cook until the edges begin to turn a golden brown. Add all the sliced peppers into the pan and stir. Season with salt and pepper. Lower the heat to medium and cook the pepper mixture for 10 minutes. Once the peppers are limp, add the thyme and chicken stock and simmer uncovered until most of the stock has evaporated and the pan is almost dry. Add 2 or 3 drops of red wine vinegar. Keep the stew warm on the side until ready to serve.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.