Melt the butter and vegetable oil in a 12-inch nonstick skillet over high heat until the butter is melted and bubbling.

Stir in the brown sugar and salt, then add the onions and stir to coat them thoroughly with the fat.

Cook the onions, stirring occasionally, until they begin to soften and release their moisture, about 5 minutes.

Reduce the heat to medium and continue cooking, stirring often, until the onions are deeply browned and slightly sticky, about 35 to 40 minutes. If they start to scorch, reduce the heat, or increase it slightly if they haven't begun browning after 20 minutes.

Remove the pan from the heat, stir in the water and pepper, then transfer the onions to a large plate to cool to room temperature.

Stretch the pizza dough onto a prepared baking sheet or pizza stone and bake until the surface is just starting to turn spotty brown, about 15 to 17 minutes.

Spread the cooled caramelized onions evenly over the surface of the par-baked pizza and sprinkle with the shredded Gruyère and minced thyme.

Return the pizza to the oven and bake until the cheese is melted and beginning to brown in spots, 5 to 10 minutes longer. Serve hot, while the cheese is still bubbly.