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1.
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In a large skillet, melt butter over medium-high heat.
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2.
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Add shallot and mushrooms into the butter and give the mixture a good stir.
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3.
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Stir brown sugar, salt, pepper, and the leaves of 1 thyme sprig into the mushroom mixture.
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Stir to combine and cook until mushrooms and shallots have caramelized, about 5 minutes.
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Remove cooked mushroom mixture from heat and set aside.
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4.
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Heat a cast iron grill pan on the stove over medium heat.
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If you dont have a grill pan, you can use a real, outdoor grill (lucky you) or just bake the flatbread in the oven at 400 F like you would a regular pizza.
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5.
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Roll out the pizza dough to about 1/4 inch thick.
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Depending on the size of your pan, you may have to make smaller flatbreads and cook in batches.
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6.
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Press dough onto the hot grill pan.
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I like to use tongs to press the corners down so that I dont burn my fingers.
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Let dough cook for 2-3 minutes, until you can easily pick it up with tongs and dough has dark, charred grill marks.
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7.
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While the dough cooks, pour balsamic vinegar into a small skillet and heat over medium-high heat until it reduces to a thick syrup, about 8 minutes.
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It should be reduced to this syrup by the time the flatbread is done cooking.
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8.
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When the first side of the dough has cooked, flip the dough so the uncooked side is facing the pan.
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9.
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Drizzle olive oil over the top of the flatbread and quickly top with the mushroom mixture and crumbled goat cheese.
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10.
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Cook until the goat cheese has softened and the bottom of the flatbread has nice, charred grill marks, another 3-4 minutes.
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11.
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Remove pizza from grill pan and place on a cutting board.
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Top with remaining thyme leaves and a drizzle of balsamic glaze.
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Slice and serve immediately.
