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