Preheat the oven to 425°F (220°C). Spray a 9-inch springform pan with nonstick spray and optionally wrap the outside bottom edge with foil to prevent leaks.
Bring a large pot of salted water to a boil and cook the bucatini according to package directions, reducing the cooking time by 1–2 minutes so the pasta is al dente.
Drain the pasta in a colander and let it cool slightly. Toss briefly with a little olive oil if needed to prevent sticking.
In a large bowl, whisk together the milk, grated Parmigiano-Reggiano, beaten eggs, minced garlic, kosher salt, and ground black pepper until smooth.
Stir in 1 cup of the grated provolone and half of the grated mozzarella, then add 1/2 cup of the pesto and mix until combined.
Add the cooked bucatini to the bowl and toss thoroughly so the pasta is evenly coated with the cheese-milk mixture.
Transfer the pasta mixture to the prepared springform pan, pressing gently to even the top. Sprinkle the remaining provolone and mozzarella over the top and drizzle additional pesto as desired.
Bake for 35 to 45 minutes, or until the cheese is golden and bubbly and the center is set.
Remove from the oven and let the pie cool 10 to 15 minutes. Run a knife around the edge if needed, release the springform, slice into 6 to 8 wedges, and serve with extra pesto and fresh basil.