How to Make It
Line a baking sheet with parchment paper. In a food processor, pulse the pine nuts with the garlic until chopped. Add the basil, Parmigiano and pecorino and pulse to combine. With the machine on, drizzle in the olive oil. Season the pesto with salt and scrape into a large bowl.
In a large saucepan of salted boiling water, cook the pasta until al dente; drain well. Add the pasta and butter to the pesto, season with salt and pepper and toss well. Spread the pasta on the prepared baking sheet, cover loosely with plastic wrap and refrigerate until cold, about 1 hour.
Using your hands, form the pasta into 18 balls. Make an indentation in each ball and fill with a piece of stracchino; shape the pasta around the cheese to enclose it. Cover the croquettes with plastic wrap and refrigerate until cold, at least 3 hours or overnight.
In a large enameled cast-iron casserole, heat 3 inches of canola oil to 350°. Set a rack over a baking sheet. In a medium bowl, beat the eggs. Place the breadcrumbs in a second bowl. Season the eggs and breadcrumbs with salt. Dip each croquette in the eggs, letting the excess drip off, then roll in breadcrumbs; repeat the dipping and rolling to make a double layer of coating. Transfer the croquettes to the rack.
In 2 batches, fry the croquettes, turning, until deep golden and heated through, about 5 minutes per batch. Serve hot.
Look for trofie pasta at Italian markets and at specialty food shops.