Heat the oil in a large skillet or stockpot over medium heat. Add the onions and a generous seasoning of salt and pepper. Toss to coat well, then cover the pan. Cook, tossing occasionally, until the onions are very soft and are caramel brown all over. After the first 10 minutes or so of cooking, you will have to lower the heat to prevent sticking. The onions need to be cooked slowly over low heat in order to caramelize properly; this will take about 45 minutes. By cooking them covered, you can use less oil to achieve this. Remove the cover from the pan, then add the walnuts. Cook 5 minutes, tossing frequently. Remove the onions from the heat and let cool.
To make the crust, preheat the oven to 450 degrees. Combine the flour, baking powder, and salt in a large bowl and mix thoroughly. Pour in the beer and mix well. The dough will be sticky. Spread a handful of flour on your work surface and dump the dough onto it. Toss the dough around to coat it with the flour and prevent it from sticking. Knead it 2 or 3 times to make it pliable. Shape the dough into a ball, then divide it in two.
Grease 2 baking sheets. Use a rolling pin to roll each ball into an 11 or 12 inch circle. Spread half the onion mixture onto each pizza. Sprinkle the crumbled goat cheese all over each pizza. I like to cook one pizza at a time, but you can cook the pizzas on 2 different oven racks and alternate them halfway through the cooking. Bake 12 to 15 minutes, or until golden brown on top or underneath