For the blossoms: Heat the oil to 375 degrees F in a heavy-bottomed pot over medium-high heat.
Combine the goat cheese, Parmesan, parsley and chives in a medium bowl. Season with salt. Spoon the mixture into a piping bag; cut a hole in the bag. Gently pipe the mixture into the zucchini blossoms and pinch the ends together a bit.
Whisk together the flour and white wine in a medium bowl. Working in batches so you don¿t crowd the pot, dip each blossom into the batter and then carefully place in the hot oil. Cook until lightly puffed and golden, about 3 minutes. Remove the blossoms from the oil and drain on paper towels. Season with salt immediately.
For the noodles: Coat a large skillet with some olive oil, add the anchovy, garlic, oregano and some red pepper flakes, and turn the heat to medium. When the garlic gets brown and aromatic, remove it from the skillet and discard. Add the tomatoes, some salt, and 1/4 cup water. Cook, shaking the pan occasionally, until the water is reduced by about half.
Remove the skillet from the heat, add the zucchini noodles and toss until just wilted. Taste and season.
Toss the arugula with some olive oil and salt and divide it among 4 dinner plates. Top the arugula with some zucchini pasta and 2 zucchini blossoms, shave some Parmesan over the tops, and serve.