In a small bowl, combine warm water and warm milk (temperature ~105–115°F / 40–46°C). Stir in granulated sugar and sprinkle the instant yeast over the surface. Let sit 5–7 minutes until foamy. In a large mixing bowl, whisk together flour and fine sea salt.

Make dough

Once yeast is foamy, add melted butter to the yeast mixture, then pour into the flour. Stir with a wooden spoon or dough hook until a shaggy dough forms. If using a stand mixer, mix on low with dough hook for 3–4 minutes until cohesive; otherwise, turn onto a lightly floured surface and knead by hand 6–8 minutes until smooth and elastic. The dough should be soft and slightly tacky but not sticky; add up to 2 tablespoons extra flour if too sticky.

First rise

Lightly oil a clean bowl with the vegetable oil, place the dough in the bowl, turning to coat. Cover tightly with plastic wrap or a damp towel and let rise in a warm draft-free place for 60–75 minutes, or until doubled in size.

Garlic Butter Topping

Prepare garlic butter

While the dough rises, melt 4 tablespoons unsalted butter in a small saucepan over low heat or in the microwave. Add minced garlic and cook gently 1 minute just to soften the raw edge — do not brown. Remove from heat and stir in chopped parsley, grated Parmesan, fine sea salt, black pepper, and set aside to cool slightly.

Shaping & Bake

Preheat and punch down

Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper. Gently punch down the risen dough to release gas and transfer to a lightly floured surface.

Shape garlic sticks

Divide the dough into 12 equal pieces (about 2 ounces / 55 g each). Roll each piece into a tight 6–8 inch stick by flattening slightly then rolling under your palms to form a tapered baton. Place sticks on the prepared baking sheet spaced about 1 inch apart.

Brush with garlic butter and egg wash

Reserve about 2 tablespoons of the prepared garlic butter for serving. Use a pastry brush to liberally brush each stick with most of the remaining garlic butter. In a small bowl whisk the egg with 1 teaspoon water to make an egg wash; brush lightly over each stick for color. If desired, sprinkle sesame seeds or additional grated Parmesan over the sticks. Sprinkle coarse sea salt or flaky salt lightly over the tops.

Bake

Bake in the preheated oven for 12–15 minutes, rotating the sheet halfway through, until golden brown and sound hollow when tapped. Baking time can vary slightly depending on oven and size of sticks.

Finish and serve

Remove from oven and immediately brush once more with the reserved garlic butter for extra flavor and shine. Let cool 5 minutes on the sheet, then transfer to a wire rack. Serve warm with additional grated Parmesan or marinara for dipping, if desired.