Place water, yeast, and sugar in a large glass measuring cup and whisk to combine. Let stand 5 minutes or until foamy on top. Stir in olive oil.
Place flour in a large bowl and add salt.
When yeast mixture is ready, pour it into bowl with flour. Mix together using a wooden spoon until it forms a ball of dough.
Place dough on a lightly floured surface. Dough will be raggedy. To knead dough, push down on dough with heel of hand, turn dough 1/4 turn, fold it by taking the edge closest to you and pulling it up and over to the far edge. Continue to knead for 3 to 5 minutes of until dough is elastic and smooth. If dough begins to stick, lightly dust surface with flour. Dough should be just a little sticky.
Place dough in a bowl that has been coated with olive oil. Turn dough to coat in olive oil. Cover bowl with plastic wrap and let rise at room temperature for 1 1/2 to 2 hours. It should double in size.
While dough rises, make pimento cheese and saute the onions. In a small bowl, mix together cheddar cheese, pimentos, and mayonnaise. Just use enough mayo to bind it together. If you use too much, it will make the focaccia greasy. To make onions, saute them in 1 tablespoon of olive oil. Sprinkle them with a pinch of sugar and 1/4 teaspoon of salt just after placing them in the pan. Cook until golden brown.
Coat a sheet pan with a liberal amount of olive oil. Place dough on pan. Spread dough out with your hands, shaping it to fit the pan. If it is retracting back a lot, cover it with a dish towel and let it rest 5-10 minutes before resuming. Once you have fully shaped the dough, use your fingertips to press dimples all over the dough, then cover it with a towel and let it rise 30 minutes.
Place a baking stone in the oven and preheat oven to 450 degrees for 30 minutes.
Once dough has risen the second time, use your fingertips to press more dimples and then drizzle liberally with olive oil and sprinkle lightly with kosher salt.
Use a spoon to place dollops of pimento cheese on the dough and scatter caramelized onions and pulled pork evenly across dough. Bake for 20 to 25 minutes.