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1
Whisk the water and sugar together in the bowl of a stand mixer fitted with a dough hook, then sprinkle the yeast on top.
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2
Set aside until the mixture bubbles, about 10 minutes.
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3
Add the flour, measured oil, and salt and mix on low speed until the dough just starts to come together and looks shaggy, about 2 minutes.
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4
Increase the speed to medium and continue to mix until the dough is elastic and smooth, about 5 minutes.
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5
Meanwhile, coat a large bowl with a thin layer of oil and set it aside.
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6
When the dough is ready, form it into a ball, place it in the oiled bowl, and turn to coat it with the oil.
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7
Cover the bowl tightly with plastic wrap.
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8
Let the dough rest in a warm place until it doubles in size, about 1 hour.
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9
Meanwhile, make the sauce.
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10
Place all of the ingredients in a medium saucepan and bring to a simmer over medium heat, whisking occasionally.
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11
Reduce the heat to low and continue to simmer, whisking occasionally, until the flavors have mingled and the sauce has reduced to about 1 cup, about 30 minutes.
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12
Remove the pan from the heat and let the sauce cool to room temperature.
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13
Once the dough has risen, punch it down, turn it out onto a work surface, and form it into a ball.
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14
Return the dough to the bowl, cover tightly with plastic wrap, and let rest for 15 to 20 minutes.
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15
Meanwhile, heat the oven to 525 degrees F and arrange a rack on the bottom.
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16
Transfer the dough to a baking sheet.
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17
Using your hands, stretch and shape it into a rough 16-by-12-inch rectangle.
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18
Using a fork, prick the dough all over, being careful not to poke all the way through.
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19
Evenly spread all of the cooled barbecue sauce over the dough, leaving about a 1/2-inch border around the edge.
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20
Evenly scatter the chicken over the sauce, sprinkle both cheeses over the chicken, and scatter the onion over the cheese.
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21
Bake until the crust is golden brown and the cheese has melted, about 20 minutes.
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22
Transfer to a cutting board, cut into slices, and serve.
