Bring 4 quarts of water to boil in a large pot. While water is coming to a boil, heat 1 tablespoon oil in a 10-inch skillet over medium heat until shimmering. Add panko and cook, stirring frequently, until golden brown, about 4 minutes. Off heat, stir in ⅛ teaspoon salt and ⅛ teaspoon pepper. Transfer panko to a plate to cool.

Add broccoli stalks, garlic, and 1 tablespoon salt to boiling water and cook, stirring occasionally, until stalks are tender, about 6 minutes. Using a spider skimmer or slotted spoon, transfer stalks and garlic to blender; set aside.

Return water to boil. Add florets and cook until bright green and tender, about 3 minutes. Using a spider skimmer, transfer two-thirds of the florets to a bowl. Transfer remaining florets to blender with stalks and garlic. Return water to boil, add pasta, and cook, stirring often, until al dente.

While pasta cooks, add basil; yogurt; Pecorino; pine nuts; lemon zest and juice; anchovies, if using; and remaining ½ cup oil, 2 teaspoons salt, and 1 teaspoon pepper to blender and process until mixture resembles pesto, about 20 seconds, scraping down sides of blender jar.

With blender running, slowly add ½ cup water and process until sauce is smooth and thick but fluid, about 20 seconds, adding additional water as needed.

Reserve 1 cup of the pasta cooking water, then drain the pasta and return it to the pot. Off heat, add sauce and reserved florets, and stir gently to combine. Adjust consistency with reserved cooking water as needed. Season with salt to taste, and serve immediately, passing panko and extra Pecorino separately.