Lightly grease a half-sheet pan (13″ x 18″) or other large, rimmed baking sheet. Drizzle with 1 tablespoon olive oil; set it aside.
Bread machine instructions: Add ingredients in order listed to the bread machine. Set to dough cycle and let it work its magic, then skip to step 4.
Electric mixer or by hand instructions: Mix together all ingredients and knead or use the dough hook until a slightly soft dough forms. Cover and let rise for 1 hour, until puffy.
Spread the dough into the pan with oiled hands, stretching it to the edges. If it shrinks back, let it rest for 10 minutes, and stretch again, repeating if necessary.
Use a lightly greased bench knife or pizza wheel to cut the dough into 12 or 13 crosswise strips, 1″ wide. Then cut the dough in half lengthwise, you’ll have 24 to 26 strips, each about 6 1/2″ long. Leave them right as they are, in the pan.
Cover and let the strips rise at room temperature for 45 to 60 minutes, until puffy. Towards the end of the rising time, preheat the oven to 350°F.
Brush the strips with the egg wash, and sprinkle on the topping.
Bake the breadsticks for 30 to 35 minutes, until they’re light gold. Remove the pan from the oven.
Once the breadsticks are cool enough to handle, cut along their score lines.