To make the pizza dough, in a bowl, whisk together the water, yeast, and sugar. Let stand until foamy, about 5 minutes. Add the flour, oil, 1 teaspoon salt, and 1/2 teaspoon pepper. Stir until the dough pulls away from the sides of the bowl. Pull the dough out onto a floured work surface and knead a few times, then form into a ball. Oil a second bowl, put the ball in the bowl, and turn to coat it with oil. Cover the bowl with plastic wrap and set aside in a warm, draft-free area. Let the dough rise until doubled in size, 11 1/2 hours.
While the dough is rising, in a sauté pan over medium heat, melt the butter. Add the onions and stir to coat. Sprinkle the sugar over the onions and stir once more. Reduce the heat to medium-low and cook, stirring, until the onions are golden brown, about 20 minutes. Transfer to a bowl.
At the same time, in a second sauté pan over medium heat, cook the bacon, stirring, until crisp, about 8 minutes. Transfer to paper towels to drain.
Prepare a charcoal or gas grill for direct grilling over medium-high heat. Brush and oil the grill grate.
Dump the dough onto a floured work surface, then divide in half. Using a floured rolling pin, roll out 1 half into a round 10—12 inches (25—30 cm) in diameter and about 1/8 inch (3 mm) thick. Repeat with the second half.
Slide 1 round onto a baker's peel or a rimless baking sheet dusted with cornmeal, and then carefully slide the round off the peel onto the grill directly over the fire. Repeat with the second round. Cook the crusts until the underside is well marked, about 2 minutes. Using the peel, transfer the crusts, grilled side up, back to the work surface.
Top each crust with half the onions, half the bacon, and then half the cheese. Season with salt and pepper. Using the peel, return the topped crusts to the grill, cover the grill, and cook until each crust is firm and browned at the edges and the cheese has melted a bit, 4—6 minutes.
Transfer the pizzas to a cutting board and sprinkle with the thyme. Cut into wedges and serve at once.