For the sponge:
Place 1 cup water and yeast in the bowl of a stand mixer. Stir well with a spatula and let stand for about two minutes. Add the 2 cups bread flour. Mix until well combined (alternately, mix by hand with a wooden spoon or spatula). Cover bowl and let rest in a warm place until doubled in size, about an hour and a half.
For the dough:
Add 2/3 cups water, olive oil and salt to the bowl with the sponge in it. Stir until the sponge is broken up a bit and the salt is dissolved.
Add the additional 2 cups of bread flour. Mix in a stand mixer fitted with the dough hook on medium speed for 1 minute, then use a spatula or a bowl scraper to scrape the bottom of the bowl to make sure all the flour is incorporated. Mix about 8-9 minutes more, until the dough is a cohesive ball and pulls away from the sides of the bowl and most of the bottom.
Cover bowl and let rest in a warm spot for 1 hour, until risen significantly. Gently fold the dough over itself a few times and press down to deflate slightly. Cover bowl and let rest again for about 30 minutes, until the dough has again risen significantly. It should be airy and full of bubbles.
For the filling and assembly:
While the dough is undergoing the second rise, prepare the garlic knot filling. Place the garlic and butter in a small saucepan set over very low heat. Let the butter melt and poach the garlic for about 20 minutes, keeping the temperature of the mixture very low so that the butter does not burn. If the butter starts to bubble vigorously, turn off the heat and let the residual heat poach the garlic. Remove saucepan from the heat and allow mixture to cool.
Preheat oven to 375°F. Brush the base of a Bread Oven with some of the melted garlic butter from the saucepan.
Scoop the dough out of the bowl and onto a clean, floured work surface. Pat the dough into a rectangle. Then using a rolling pin, roll dough out to about 18 x 13 inches, with the short edge facing you.
Use a pastry brush or spoon to spread the poached garlic over half of the dough closest to you, leaving about an inch around the edges. Sprinkle the parsley and 1/2 cup of parmesan over the garlic butter (still leaving the other half of the dough clear).
Lift up the top half (without the garlic butter and cheese) and fold it over the half closest to you to cover the filling. Pinch the dough closed a little around the edges. Sprinkle a little more flour on top and underneath the dough. Roll dough out again to about 18 x 13 inches.
Use a kitchen knife or a dough knife to cut off thin strips of the dough, about 1 to 2-inches wide. Wrap the dough around two fingers a few times and tie a simple overhand knot, place in the base of the Bread Oven. Repeat with the remainder of the dough, nestling the knots close together in the pan.
Once all the knots are shaped, cover Bread Oven with the lid and let rise in a warm spot until puffy, about 20 minutes. Remove lid and drizzle any remaining melted butter over the top of the dough.
Place pan in preheated oven and bake uncovered until the bread is golden brown on top, about 25 minutes. Remove from the oven and immediately sprinkle with remaining parmesan cheese. Let cool slightly before serving.