Combine the sourdough starter, warm water, active dry yeast, salt, sugar, and olive oil in a mixing bowl. The mixture should look loose and evenly blended before any flour goes in.
Mix with a dough hook and add the flour gradually as the dough starts to come together. It will move from shaggy and wet to a soft mass that still looks a little sticky.
Keep adding flour until you have a soft, sticky dough rather than a stiff one. The dough should cling slightly to the bowl but still gather around the hook.
Flour your work surface well, then turn the dough out and dust the top. This extra flour keeps the dough from welding itself to the counter.
Lightly flour your hands before kneading the dough. That small step makes the dough much easier to handle without tearing.
Knead until the dough feels soft, smooth, and no longer sticks to your hands. When it is ready, it should feel tacky but not messy and spring back a bit when pressed.
Set the dough in a well-oiled dish, cover it, and let it rise. Look for visible expansion and a puffier, airier feel before moving on.
Preheat the oven to 375F. Getting the oven ready now helps the shaped bombs bake evenly without waiting around.
Press the risen dough down, then knead it briefly again to form a smooth ball. This redistributes the air and makes the dough easier to portion evenly.
Divide the dough into 24 pieces. Even portions help the pizza bombs bake at the same pace and finish with a more uniform shape.
Take one piece, knead it two or three times, roll it into a ball, and flatten it. You want a compact round that can hold filling without becoming too thin at the center.
Add the filling with cheese first, then your toppings. Keeping the cheese underneath helps cushion the dough from wetter toppings.
Roll the filled dough up, fold it in half, and place it in your baking pan. Muffin tins make more distinct bombs, while a 9x13 dish gives you a pull-apart bake.
Repeat with the remaining dough pieces until all 24 are shaped and in the pan. Try to keep the filled pieces similar in size so none overbake while others lag behind.
Bake for 15 minutes. The dough should start setting, puffing, and taking on light color around the edges.
Brush the tops with melted butter and sprinkle with dried oregano. This is where the rich toasted aroma starts showing up in a big way.
Return the pan to the oven for another 5 minutes. The tops should finish golden brown and the crust should look dry and baked rather than pale.
If you used a muffin pan, remove the pizza bombs and cool them on a rack. This keeps the bottoms from trapping steam and turning soft.
Serve warm with warm pizza sauce for dipping. The centers should be melty and the crust should pull apart easily while still holding the filling.