Combine flour, 1/4 teaspoon salt, olive oil, and dry white wine in a stand mixer. Mix using the paddle attachment until a dough ball forms. Switch to the dough hook and knead the dough for 10 minutes.

Line a baking sheet with parchment paper. Bring a large pot of water to boil over high heat and add the remaining salt. Spread out a clean cloth on a counter or table.

Pinch off a walnut-sized piece of dough. Roll into a 2 1/2-inch snake and form a tear drop shape, lightly pressing the ends together to seal into a ring. Repeat with remaining dough.

Turn the boiling water to a low boil. Drop taralli a few at a time into water. Let them sink and float on their own. Remove floating taralli with a slotted spoon to drain on the clean cloth. Repeat for all remaining dough.

Preheat the oven to 350°F (175°C).