Prepare the Pasta Heat butter and oil in a large Dutch oven over medium. Add onions and ½ teaspoon of the salt; stir onions until evenly coated in butter. Cover and cook, stirring occasionally, until onions begin to soften, about 10 minutes. Uncover, and continue cooking over medium, stirring occasionally and scraping browned bits from bottom of pan, until onions are deep golden brown, meltingly soft, and jammy, 50 minutes to 1 hour. Remove ½ cup of the caramelized onions, and place in a small bowl; set aside.

While onions cook, prepare the Gruyère Breadcrumbs Melt butter in a large skillet over medium. Stir in panko, thyme, garlic, salt, and pepper. Cook, stirring constantly, until panko is toasted and golden brown, 4 to 5 minutes. Transfer panko mixture to a medium bowl; set aside to cool until ready to use.

When onions are nearly done caramelizing, bring a large pot of salted water to a boil; add pasta, and cook to al dente according to package directions. Drain pasta, reserving ½ cup cooking water.

Once onions are caramelized, stir in sherry, scraping bottom of Dutch oven to loosen and dissolve browned bits. Cook, stirring constantly, until sherry has evaporated and alcohol smell has dissipated, about 1 minute. Stir in flour until onions are evenly coated. Add broth, Worcestershire sauce, fish sauce, pepper, thyme, bay leaf, and remaining 1 ½ teaspoons salt. Bring to a simmer, stirring often. Continue to simmer, stirring occasionally, until liquid is slightly reduced and thickened, about 5 minutes. Remove and discard thyme sprigs and bay leaf.

Remove Dutch oven from heat; gradually stir in Parmigiano-Reggiano and shredded Gruyère, one handful at a time, until cheeses are melted. Add drained pasta and ¼ cup of the reserved cooking water; toss until well combined. Continue tossing until a glossy sauce forms that evenly coats pasta, gradually adding remaining ¼ cup cooking water, 1 tablespoon at a time, if needed. Stir in reserved caramelized onions until evenly distributed.

Divide pasta evenly between 4 bowls. Stir grated Gruyère into Breadcrumbs, and sprinkle evenly over pasta. Garnish with additional thyme leaves and grated Gruyère. Serve immediately.