Whisk 2/3 cup warm water (105 degrees F) and the brown sugar together in a small bowl; sprinkle the yeast on top. Set aside until foamy, about 5 minutes. Stir in 1 tablespoon of the olive oil.
Whisk the flour and 1 teaspoon salt in a large bowl. Make a well in the center of the flour and pour in the yeast mixture. Gradually stir with a wooden spoon to make a rough dough.
Turn the dough out onto a lightly floured surface. Knead until smooth and elastic, about 5 minutes, dusting with more flour if necessary. Form into a ball. Brush 1 large bowl with olive oil, add the dough and turn to coat in the oil. Cover tightly with plastic wrap and set aside at room temperature until doubled in size, 1 hour to 1 hour 30 minutes.
Meanwhile, add the remaining 2 tablespoons oil and half of the grated garlic to a medium saucepan. Stir to combine. Place over medium heat and cook until the oil starts to sizzle, about 2 minutes. Add the marinara sauce and crushed red chile flakes. Stir to combine and bring the sauce to a simmer. Continue to cook for 10 minutes to allow the flavors to develop. Turn off the heat and stir in the basil leaves. Season to taste with additional salt if needed then cover with a lid to keep warm.
Turn the risen dough out onto a lightly floured surface. Cut the dough into 4 equal pieces. Flatten each piece into a 4-inch square. Cut each square into 4 strips then tie each strip into a knot. Transfer to a baking sheet lined with parchment paper, making sure there is enough room between each knot to allow it to almost double in size. Cover with a clean kitchen towel and set in a warm place to rise until the knots almost double in size, about 30 minutes.
Position a rack in the center of the oven and preheat to 425 degrees F.
Line 1 baking sheet with a dry kitchen towel and another baking sheet with parchment paper sprayed with nonstick spray.
Bring 8 cups of water to a boil in a deep straight-sided skillet. Stir in the baking soda. Add half of the knots and cook until lightly puffed, about 30 seconds, flipping once halfway through the cooking time. Transfer to the towel-lined baking sheet using a slotted spoon. Repeat with the remaining knots. Let the knots dry for 1 minute before transferring to the parchment-lined baking sheet.
Whisk the egg yolk with 1 tablespoon water together in a cup. Brush the mixture over the knots. Sprinkle the tops with additional salt.
Bake the knots until deep golden brown and cooked through, 15 to 18 minutes, rotating the pan once halfway through the cooking time.
Cool the knots slightly while you make the garlic butter. Add the butter to a small saucepan and cook over medium heat until it has melted and starts to bubble, 3 to 4 minutes. Stir in the remaining grated garlic and a pinch of salt and cook for 2 minutes, stirring constantly. Pour the butter into a large bowl and add the garlic knots. Use a rubber spatula to stir and coat the knots in the garlic butter, stirring for about 30 seconds to allow the butter to mostly absorb into the knots. Sprinkle in the parsley and stir to combine. Sprinkle in the Parmesan and stir until just coated. Transfer to a large platter.
Serve the knots with the warm marinara sauce on the side.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)