Dough: In the bowl of a stand mixer fitted with the paddle attachment on low speed, sift together 1 1/2 cups flour, sugar, salt and yeast.
In a small saucepan melt butter over medium heat. Add water and yogurt, increase heat to medium high. Cook for 2-3 minutes stirring constantly just until the mixture reaches 120°F-130°F. Remove from heat, add vinegar, mix well.
Add warm liquid to sifted dry ingredients in the bowl. Beat with electric mixer fitted with the paddle attachment on medium speed just until smooth.
Lower the speed of the mixer, switch to the dough hook. Gradually add 3 1/2 cups flour. Knead on medium speed just until dough isn't sticky and easy to handle, about 5 minutes. Add additional flour 1 tablespoon at a time (up to 1/2 cup total) only if sticky. The dough should pull away from the sides.
Turn dough onto a lightly floured surface and turn to coat lightly with flour, kneading just until smooth and elastic. Grease mixing bowl, add dough back to bowl and cover with a damp towel. Let rise 45-60 minutes or until doubled in size.
Breadsticks: Divide dough in half. Roll each half into a 10 x 8-inch rectangle. Use a sharp knife to cut each rectangle into 8 (1-inch wide) strips. Stretch each strip to about 12 inches long, fold in half and twist several times, pinching both ends to seal. Place onto a lightly buttered baking sheet around 1-2 inches apart. Repeat with remaining dough strips.
Cover pan loosely with plastic wrap spritzed with cooking spray. Let rise in a warm draft free place for15-20 minutes or just until doubled in size.
Bake: Preheat oven to 375°F. Mix together melted butter, granulated garlic and Italian seasoning in a small bowl.
Uncover breadsticks, brush the tops and sides with melted butter mixture and sprinkle with Parmesan cheese.
Bake for 20-25 minutes or until golden. Serve warm or at room temperature brushed with additional melted butter, if desired.