In a large bowl, combine the yeast, sugar and 1 cup lukewarm water (105 to 115° F). Let stand for 5 minutes until bubbly.
Stir 1 1/2 cups flour, 2 tablespoons olive oil and the table salt into the yeast mixture. Stir together with a wooden spoon or beat with an electric mixer until the dough is shiny and elastic, 3 to 5 minutes. Add the remaining 1 cup flour and mix until incorporated. Turn the dough out onto a lightly floured work surface, cover with an inverted bowl and let stand for 15 minutes. On the lightly floured work surface, knead the dough until smooth and springy, 5 to 10 minutes. Place the dough in a large bowl, cover with plastic wrap and let rise until puffy, about 45 minutes.
Preheat the oven to 350°. Drizzle the remaining 4 tablespoons olive oil over 2 baking sheets. Divide the dough into 4 equal pieces, then cut each piece again into 4 equal pieces to make a total of 16 pieces. On a lightly floured work surface, using floured hands, stretch and roll each piece of dough into a 15-inch-long breadstick. Place the breadsticks about 1/2 inch apart on the prepared baking sheets. Roll the breadsticks in the olive oil on the baking sheets to coat and sprinkle with the coarse salt. Bake until golden and crisp, about 30 minutes. Transfer the baking sheets to a rack and let the breadsticks cool completely.