In a large bowl, combine the bread flour, yeast and lukewarm water. Turn out onto a floured surface and knead for 10 minutes, or until smooth and elastic. Place in a lightly oiled bowl, cover with plastic wrap and allow to proof for at least 30 minutes, or until doubled in size.
Prepare the barbecue for indirect cooking over medium- high heat (210°C to 240°C) with a pizza stone. If using a Weber Q barbecue, set up your barbecue with a convection tray and a trivet. Preheat a large Weber pizza stone for 20 minutes.
Once the dough has risen, turn the dough out onto a floured surface. Divide into two and roll into two 30cm round bases. Drizzle 2 tablespoons of olive oil onto two large Weber pizza serving trays. Place the bases onto the trays. Top each base with the pizza sauce, bacon, baked beans, cherry tomatoes and cheese. Using the back of a spoon make 3 indentations into the toppings of each pizza, creating a space to crack the eggs into.
Roast the pizzas, one at a time over indirect medium-high heat for 25 minutes. 5 minutes into the cooking time, crack the eggs onto the pizza, inside the indentations. Continue to cook the pizzas for a further 20 minutes, or until the base is cooked and the cheese is golden.
Remove the pizzas from the barbecue with heatproof gloves. Season with salt and pepper and garnish with fresh basil.