Preheat your oven to 425°F (220°C). Place a baking sheet in the oven to warm up.
In a medium skillet, melt the butter over medium heat. Add the sliced pears and sauté for 3-4 minutes until they start to soften.
Sprinkle the brown sugar over the pears and stir gently. Continue to cook for another 5-7 minutes until the pears are caramelized and golden. Remove from heat and set aside.
On a lightly floured surface, stretch or roll out the flatbread to your desired thickness.
Carefully remove the preheated baking sheet from the oven and place the flatbread on it.
Drizzle a little olive oil over the flatbread and season with salt and pepper. Spread the caramelized pears evenly over the flatbread.
Sprinkle the crumbled goat cheese and chopped walnuts over the top of the pears.
Sprinkle fresh thyme leaves over the flatbread (reserve a few for garnish if desired).
Bake in the oven for about 10-12 minutes, or until the flatbread is golden and the cheese is slightly bubbly.
Remove from the oven and drizzle with balsamic glaze if using. Garnish with reserved thyme leaves.