Stir water and sugar in a large bowl until the sugar is dissolved.
Add yeast and let stand for 10 minutes; it should bubble (if it doesn't, throw it out and start over with fresh yeast).
Add whole-wheat flour, 1 3/4 cups all-purpose flour, 2 tablespoons oil and 1 teaspoon salt. Mix with a wooden spoon and, eventually, your hands until combined. Add up to 1/4 cup more all-purpose flour if the dough seems sticky.
Turn the dough out onto a lightly floured surface and knead until it is smooth and elastic, at least 5 minutes.
Lightly oil a clean large bowl.
Gather the dough into a ball; place it in the bowl and turn to coat. Cover with a clean kitchen towel and set aside in a warm, draft-free place until doubled in size, 1 to 1 1/2 hours.
Position racks in the upper and lower thirds of oven; preheat to 375°F. Coat 2 large baking sheets with cooking spray.
Cut the dough into 12 equal pieces. Stretch and roll each piece to about 24 inches long and approximately 3/8 inch thick. Cut each piece in half to make 2 dozen 12-inch-long breadsticks. Place about 1/2 inch apart on the prepared baking sheets. Brush both sides of the breadsticks with the remaining 2 tablespoons oil and sprinkle with the remaining 1/2 teaspoon salt.
Bake the breadsticks for 15 minutes. Gently turn each breadstick over. Continue baking until crisp and golden brown, 15 to 20 minutes more.