Warm the milk to 110 degrees F and pour into a mixing bowl. Dissolve the yeast in the milk in a mixing bowl. Add in the flour and the sugar and mix. Let sit until the mixture (sponge) begins to bubble, about 6 to 8 minutes.
Add in the flour, salt, cold water, and olive oil to the sponge mixture. Blend until a dough is formed. When mixed well, the dough will begin to pull away from the bowl. Knead lightly until a large ball is formed. Cover with a towel and let sit. When the dough has risen about 30 to 45 minutes, punch down, and begin to roll into 6-ounce pieces for folded pizzas and 4-ounce pieces for open flat bread.
Preheat oven to 500 degrees F.
In a large saute pan, melt the butter and brown lightly on medium high heat. Add in the onions, turn the heat to medium and slowly caramelize the onions. When brown, set aside to cool.
Roll out a 6-ounce piece of dough into a long rectangular shape. Place on a large pizza peel or baking sheet that has been sprinkled with semolina. Brush the dough with olive oil and poke it several times with a fork to keep it from bubbling up. Spread the caramelized onions evenly over the dough and sprinkle with the feta and thyme.
Place the pizza in the oven and turn heat down to 400 degrees and bake for 25 minutes or until the top is golden brown. Cut into 8 pieces and serve warm.