In a medium bowl, dissolve the sugar in the warm water. Sprinkle the yeast on top and leave for 10-15 minutes, until frothy.
In a large bowl, combine the flour and salt. Add the yeast mixture and the olive oil and mix until a soft dough forms.
Turn out the dough onto a lightly floured surface and knead for about five minutes. The dough should be a bit sticky, but you can add a little more flour if it's too sticky to knead. When you've finished kneading, the dough should be soft, smooth, and a little sticky.
Transfer dough to a large, lightly oiled bowl and turn to coat all over. Cover and put in a warm place for about an hour, until doubled in size.
Towards the end of the rising time, start preheating the oven to 400°F (200°C). Line two large baking sheets with parchment paper or silicone baking mats.
Melt the butter in the microwave and add the salt and garlic; stir to combine.
Turn out the dough onto a lightly floured work surface and knead gently to work out any large air bubbles. Roll out into a rectangle, roughly 10x12 inches, and cut into 12 1-inch strips. Fold the long strips of dough in half and twist to form the breadsticks.
Arrange breadsticks on the baking sheets. Brush generously with the butter mixture and sprinkle with Parmesan.
Bake for 12-15 minutes, until lightly browned. After about 8 minutes, rotate the baking sheets to ensure an even bake.
Brush breadsticks with the remaining butter mixture while still hot and serve warm, or freeze (see note).