Combine warm water (about 105–115°F / 40–46°C), warm milk, sugar, and instant dry yeast in a large measuring cup or bowl. Stir briefly and let sit 5–8 minutes until foamy.

In the bowl of a stand mixer fitted with a dough hook (or in a large bowl), whisk together the flour and salt. Start the mixer on low and slowly pour in the yeast mixture and the melted butter. Mix until a rough dough forms, then increase speed to medium and knead 6–8 minutes until the dough is smooth, elastic, and slightly tacky. If kneading by hand, knead on a lightly floured surface for 10–12 minutes.

Lightly oil a clean bowl with the vegetable oil, place the dough inside, turning once to coat. Cover the bowl with plastic wrap or a damp towel and let rise in a warm, draft-free place until doubled in size, about 60–90 minutes.

Punch down the risen dough and turn it out onto a lightly floured surface. Divide the dough into 12 equal pieces (about 65–75 g each). Let rest 10 minutes, covered, to relax the gluten.

Roll each piece into a 10–12 inch (25–30 cm) thin stick about 1/2 inch (1.3 cm) thick — or shape as desired for traditional garlic stick form. Place sticks on a parchment-lined baking sheet, leaving 1 inch between them. Cover loosely and let rise 20–30 minutes until slightly puffy.

Garlic Butter Topping

While the shaped dough is resting, prepare the garlic butter. In a small saucepan or microwave-safe bowl, melt the 6 tablespoons unsalted butter over low heat. Remove from heat and stir in minced garlic, chopped parsley, grated Parmesan, 1 teaspoon salt, black pepper, and red pepper flakes (if using). Taste and adjust seasoning. Keep warm.

Preheat oven to 400°F (200°C) with a rack in the center while butter is being prepared and dough is finishing its second rise.

Finishing

Prepare egg wash by whisking the egg with 1 teaspoon water. Lightly brush the tops of each risen dough stick with egg wash — this will help the garlic butter and Parmesan adhere and give a golden color.

Generously brush each stick with about 1 to 1 1/2 teaspoons of the garlic butter mixture, ensuring minced garlic is distributed. Reserve about 1 tablespoon of the garlic butter to brush again right after baking if you want an extra glossy finish.

Optionally sprinkle each stick with a light pinch of coarse or flaky sea salt for texture.

Bake the garlic sticks in the preheated 400°F (200°C) oven for 12–15 minutes, or until golden brown on top and the bottoms sound hollow when tapped. Rotate the pan halfway through baking for even color.

Remove from oven and immediately brush with the reserved garlic butter. If desired, sprinkle a little extra grated Parmesan and chopped parsley on top. Let cool 5 minutes on the pan, then transfer to a wire rack.

Serve warm with marinara or as-is. Leftover garlic sticks keep in an airtight container at room temperature for 1–2 days; reheat in a 350°F (175°C) oven for 5–7 minutes to refresh.