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1
Add water to a large bowl, then add the yeast and gently stir. Let sit for 5 minutes, until little bubbles start to appear at the top of the water. Add the flour, salt, and olive oil, and mix thoroughly with a wooden spoon until a shaggy, wet dough forms. Lightly cover a surface with about 1/3 cup flour, and pour the dough onto floured surface. Knead for 5 minutes, until dough is smooth, homogenous, and sticky. If you poke it gently, it will spring back in about 5 seconds to fill the poke mark. (Note: You can make the dough in a stand mixer with a dough hook instead, if you prefer-do the yeast dissolution in the stand mixer, and add the flour, salt, and oil, then mix with the dough hook for 3 to 4 minutes. Be sure to sprinkle in another 1/3 cup or so of flour gradually as you mix to get the same texture as you would by hand-kneading on a floured surface.)
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2
Add 3 tablespoons olive oil to a clean bowl, and swirl the bowl around to ensure the sides are greased. Form a loose ball of your dough, and transfer it to the oiled bowl, then gently flip it so both sides are covered in oil. Cover the bowl with plastic wrap and let it sit at room temperature for 60 to 90 minutes, until it roughly doubles in size.
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3
Add 1/2 cup olive oil to the bottom of a 9x13-inch cake pan. (This will feel like a lot of oil, but it gets absorbed by the dough as it cooks and makes the most amazing crust.) Transfer your dough to the pan and gently stretch it to fit the dimensions. It will feel like the consistency of soft chewing gum. Sprinkle 1/2 cup of green onions and a large pinch of salt across the top. On top of the green onions, use your fingers to slather the tomato puree, avoiding the edges where the dough meets pan (the sauce will burn). Loosely cover and let the dough proof in the pan for 30 minutes, until it's puffed up.
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4
Meanwhile, heat your oven to 475u00b0 F. Once the dough has puffed up, it's time for the most fun part of making focaccia: poking it all over with your fingertips to make dimples. Do this, then bake for 20 to 25 minutes, until the un-sauced edges are a toasty color where they touch the pan, and the pizza-like bubbles that have puffed up across the surface are tinged dark brown in the centers.
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5
Remove from the oven and sprinkle with the remaining green onion, a drizzle of olive oil, and more salt to taste. You can slice and eat it as soon as it cools a bit. Refrigerate to store.
