Set the oven to 400°F, cover two baking sheets with parchment, and coat the parchment with nonstick cooking spray. The greased lining will help the boiled dough release after baking.

Whisk the yeast, brown sugar, and 1/4 cup of the warm water together in the stand mixer bowl, with the dough hook ready, then leave for 5 minutes. A frothy surface signals that the yeast mixture is ready.

Stir in the flour, salt, and remaining 1 1/4 cups water, then run the mixer on low while slowly adding the melted dough butter. The mixture will gather into a firm, dense mass.

Knead at medium speed for about 7 minutes, staying beside the mixer to watch for movement on the counter. Stop when the dough feels smooth and elastic and releases cleanly from both the bowl and your hands.

Move the dough straight to a clean, unfloured counter, divide it into 10 equal portions of about 100 grams each, and roll each portion into a rope about 20 inches long. The slight grip against the counter helps lengthen the ropes without a rise before shaping.

Bend each rope into a U, cross the upper ends, turn them around one another once, then fold them down and press them onto the bottom curve before placing the pretzel on a prepared sheet. A light press should hold the ends in place while leaving the familiar open pretzel shape.

Cover the shaped pretzels and let them rise for 30 minutes while preparing the bath. They should look slightly fuller and pillowy, rather than dramatically enlarged.

Heat the bath water and baking soda in a large pot until gently boiling, then lower the risen pretzels into it in small batches for 30 seconds on each side. The hot alkaline water prepares the surface for its chewy texture and deep oven browning.

Lift each pretzel out with a slotted spoon, return it to the prepared sheet, and use a paring knife to make two cuts across its thickest section. Every pretzel should have its bath and scoring completed before it goes into the oven.

Bake both sheets at 400°F for about 16 minutes, switching their positions halfway through for more even coloring. Look for a deep golden crust, allowing slightly less baking for a softer finish or slightly more for a crisper one.

During baking, warm the topping butter in a small saucepan over medium-low heat, stir in the minced garlic for 1 to 2 minutes, then mix in the salt, garlic powder, and parsley. The garlic should smell fragrant without developing dark, bitter edges.

Coat the hot pretzels all over with the garlic butter, immediately scatter the Parmesan generously across them, and serve warm. The cheese should cling to the glossy butter coating.