Bring a large pot of water to a boil over high heat while you prepare the toasted panko and sauce.

Heat a large, high-sided skillet or braiser over medium heat. Add the panko, 2 tablespoons olive oil, and ¼ teaspoon kosher salt. Cook, stirring often, until the breadcrumbs are crisp and golden brown, 2 to 3 minutes. Transfer the toasted panko to a paper towel-lined plate and set aside. Carefully wipe the skillet clean.

Heat the remaining 2 tablespoons olive oil in the skillet over medium-high heat. Add the mushrooms and sprinkle them with the remaining ¾ teaspoon kosher salt. Cook, stirring occasionally, until the mushrooms are browned and nearly dry, 10 to 12 minutes.

Add the shallots, smashed garlic cloves, and butter. Cook, stirring often, until the shallots have softened and the mixture is fragrant, about 3 minutes. Stir in the thyme and crushed red pepper flakes.

Pour in the vermouth, scraping the bottom of the skillet to loosen the flavorful browned bits. Cook over medium-high heat until the vermouth has almost completely evaporated, about 2 minutes.

Stir in the chicken broth and bring the mixture to a simmer. Cook over medium heat, stirring occasionally, until the broth has reduced by approximately half, about 5 minutes.

Stir in the heavy cream. Remove and discard the smashed garlic cloves. Add the diced rotisserie chicken, any accumulated chicken juices, and the frozen peas. Cook over low heat for 2 to 3 minutes, or until the chicken and peas are heated through. Keep the sauce warm over low heat while the rigatoni cooks.

Before transferring the rigatoni, reserve at least 1 cup of the pasta cooking water. Using a spider or slotted spoon, transfer the molto al dente rigatoni directly into the skillet with the sauce.

Increase the heat to medium-high and add ½ cup of the reserved pasta water. Toss frequently and cook until the rigatoni is al dente and the sauce has thickened slightly, about 2 minutes. Add another splash of pasta water if the sauce becomes too thick.

Remove the skillet from the heat. Gradually add the grated Parmigiano-Reggiano, tossing continuously until the cheese melts and the sauce becomes glossy and clings to the rigatoni. Taste and add more salt if needed.

Divide the chicken tetrazzini evenly among six bowls. Finish each serving with the toasted panko and additional Parmigiano-Reggiano. Serve immediately.