Bring 3 quarts water to boil in large Dutch oven. Add 1 pound gemelli and 1 tablespoon salt and cook, stirring often, until flexible but still very firm in center, about 6 minutes. Reserve 3 cups cooking water, then drain pasta.

Meanwhile, melt 3 tablespoons unsalted butter in 12-inch nonstick skillet over medium heat. Add 2 tablespoons capers and cook, stirring constantly, until milk solids in butter are color of milk chocolate and have toasty aroma, 3 to 5 minutes. Stir in 1 cup panko and ⅛ teaspoon table salt and cook, stirring, until panko is golden brown, about 2 minutes. Transfer to bowl and stir in 2 tablespoons chopped parsley. Wipe skillet clean with paper towels, then add 1 tablespoon extra-virgin olive oil and heat over medium-high heat until shimmering. Add 2 thinly sliced Meyer lemons in even layer (slices may overlap at first but will shrink as they cook). Sprinkle with ⅛ teaspoon sugar and ⅛ teaspoon salt and cook until deeply browned on both sides, 5 to 7 minutes. Transfer lemons to cutting board and let cool for 5 minutes, then finely chop.

Melt 4 tablespoons butter in now-empty pot over medium-high heat. Stir in 1 thinly sliced shallot, 3 thinly sliced garlic cloves, and ½ teaspoon red pepper flakes and cook until softened, about 2 minutes. Stir in pasta, 2 cups reserved cooking water, and remaining ½ teaspoon salt and bring to boil. Cook, stirring often, until water is mostly absorbed but still pools slightly in bottom of pot, about 5 minutes. Add remaining 4 tablespoons butter and stir vigorously until sauce is creamy and emulsified, about 1 minute.

Off heat, stir in 1 cup grated Parmesan, ½ teaspoon pepper, and chopped lemons until incorporated. Season with salt and pepper to taste and adjust consistency with additional cooking water as needed. Serve, sprinkling with panko mixture.