In a large bowl, mix together the sugar and the warm water and stir to dissolve
Sprinkle the yeast on the water-sugar mixture. Don't mix. Allow it to stand for about 15-20 minutes until the mixture starts to bubble, indicating the yeast is alive.
Mix the flour with 1 tbsp salt. Add the water-yeast mixture one cup at a time, along with 1/3 cup of the olive oil, a teaspoon at a time, mixing between each addition either with a spoon or on low speed in a stand mixer.
Knead by hand or on low speed in a stand mixer for 10 minutes. The dough is very firm and a bit much for my ancient Hamilton Beach stand mixer to deal with because the bowl kept jumping off the stand. So I ended up kneading by hand which was a pretty good workout for my arms.
At the end you want a smooth dough. Place in a bowl coated with 1 tbsp of the oil, turning the dough over once to coat the top with oil.
Cover with a damp cloth and place in a warm place or in a cold oven, as I did, with the light on.
The dough will double in about 1 1/2 to 2 hours.
In a skillet, heat 1 tbsp of the olive oil and 1 tbsp of Earth Balance vegan butter along with 1 tsp salt.
Add the onions to the skillet and saute, over medium heat, about 25 minutes until the onions turn soft and brown. Stir frequently because you don't want the onions to burn and stick to the bottom. Set aside.
Line a baking sheet with parchment and spray or coat with oil.
Turn the dough out into the sheet and with your fingers push it into the corners to get a fairly even rectangle.
Cover and let rise for another 30 minutes.
In a small bowl, mix remaining 3 tbsp of the olive oil with the red pepper flakes, ground black pepper, rosemary and 1 tsp salt. Brush the oil mixture evenly over the top of the focaccia.
Place in a preheated, 400-degree oven and bake for 10 minutes. After 10 minutes, rotate the sheet, spread the caramelized onions over the bread, and bake another 10 minutes.
Remove the loaf to a rack and let it stand 10 minutes before cutting.