Put 3/4 cup warm water in a food processor. Sprinkle the yeast on top and let sit until foamy, about 5 minutes. Add the sugar and 1 teaspoon salt and pulse until dissolved. Add the flour, olive oil and 1 tablespoon butter and pulse until the dough forms a ball, about 1 minute. Continue to process with the motor running until smooth, about 2 more minutes.
Line 2 baking sheets with parchment; brush with olive oil. Turn the dough out onto a floured surface. Divide into 12 equal pieces, then roll each piece into a 12- to-16-inch rope. Arrange the ropes 1 1/2 inches apart on the prepared baking sheets, rolling them to coat with the oil. Cover each baking sheet with a damp cloth and set aside in a warm place until the breadsticks double in size, 45 minutes to 1 hour.
Preheat the oven to 400 degrees F. Put the remaining 3 tablespoons butter and the garlic in a microwave-safe bowl and microwave until melted, about 1 minute. Brush the breadsticks with the garlic butter; sprinkle with herbs and salt. Bake until golden, 15 to 20 minutes.