Lobster Mushroom and Butternut Squash Pappardelle

1 small butternut squash, peeled and cubed
3 tablespoons extra virgin olive oil
1 teaspoon kosher salt
500 grams dried pappardelle pasta
6 tablespoons extra virgin olive oil
1 pound fresh lobster mushrooms, cleaned and thickly sliced
4 tablespoons butter
1/2 teaspoon kosher salt
1 large shallot, finely chopped
2 cloves garlic, thinly sliced
2 salted anchovies, rinsed, bones removed and finely chopped
Pinch dried chili flakes
1 lemon, zest only
1/4 cup fresh Italian parsley leaves
1 cup Pecorino, grated
kosher salt to taste