Add the water to a large mixing bowl and sprinkle the yeast on top. Stir in the flour, salt, and 1/4 cup of olive oil. Continue stirring until a dough is formed and pulls away from the sides of the bowl. Transfer the dough to clean work surface and knead for 5 minutes.
Form the dough into a smooth ball and transfer to a large clean bowl. Drizzle the dough ball with the 2 tsp oil and flip the dough around a few times to coat in the oil. Loosely cover the bowl and set in a warm place to double in size, about 1 hour.
Preheat the oven to 425°F. Set out 3 baking sheets.
Separate dough into 6 equal size balls then separate each ball into 6 equal size balls for a total of 36 small balls. Spread the poppy seeds out in a rimmed dish and roll the dough balls in the seeds to cover, gently pushing the seeds to the dough to help them adhere. Stretch or roll each ball into long sticks roughly 10-inches long and 1/4-inch thick.
Lay the breadsticks on the baking sheets separated by at least 1-inch. Bake the breadsticks one baking sheet at a time in the oven until crispy and lightly browned, 12 to 14 minutes.
Transfer breadsticks on a wire rack to cool before serving.