Bring a large pot of water to a boil for the pasta.
Invert a small bowl inside a large bowl. Place one ear of corn upright on the top of the small bowl. Cut the kernels off the cob (they’ll fall into the big bowl); repeat with the remaining ears of corn.
In a medium bowl, pick through the crab and discard any shells; season with Old Bay.
In a large skillet, heat the oil, two turns of the pan, over medium-high. Add the bacon and render, stirring occasionally, until beginning to get crispy, 2 minutes. Add the corn and season with salt and pepper. Cook, stirring occasionally, until the corn is lightly browned, 3 to 4 minutes. Add the scallion whites, chile, thyme and garlic. Cook, stirring often, until the scallions soften, about 2 minutes. Swirl in the vermouth, then the stock; reduce the heat to low.
Salt the boiling water, add the pasta and cook until 1 minute shy of al dente. Reserve half a mug of the starchy cooking water, then drain the pasta.
Add the butter to the corn mixture; stir until melted. Add the seasoned crab and half the parsley. Add the pasta and the cooking water; toss 1 to 2 minutes to combine and season with salt.
Serve the pasta in shallow bowls topped with the scallion greens, remaining parsley and grated Parmigiano-Reggiano, if using.