Whisk the olive oil, oregano, garlic, sumac, allspice, cardamom, and brown sugar in a small bowl until the sugar dissolves and the spices are well distributed.

Heat a portion of the spice oil in a large skillet over medium heat. Add the onions, salt, and pepper and cook, stirring occasionally, until they are deeply softened and golden brown, about 20 to 25 minutes.

Heat another portion of the spice oil in a separate large skillet over medium-high heat. Add the mushrooms and cook until they have released their liquid and turned deep brown, about 10 to 12 minutes.

Adjust the oven rack to the upper-middle position, place a baking stone or inverted baking sheet on the rack, and heat the oven to 500°F.

Divide the dough into two pieces and roll each into a 15-inch round on a piece of parchment paper. Spread the onion mixture evenly over the rounds, leaving a 1/2-inch border, then top with the cooked mushrooms.

Slide the parchment with the flatbread onto the hot stone. Bake until the edges are puffed, charred in spots, and the bottom is crisp, about 8 to 10 minutes.

Toast the pine nuts in a small dry skillet over medium heat until fragrant and golden. Sprinkle the nuts and chives over the hot flatbreads and serve immediately with yogurt.