Adjust an oven rack to the upper-middle position and preheat the oven to 425°F.

Fill a large Dutch oven about three-quarter full of water and bring to a boil.  Add a generous amount of salt (about a tablespoon per gallon of water) and the noodles and cook, stirring frequently, about 3 minutes or until the noodles are just al dente (do not overcook them.)  Drain, rinse with cold water until cool, then drain again.  Set aside.

In the now-empty pot, melt 1 Tbs. butter over medium-high heat.  Add the onion and cook, stirring frequently, until softened, about 5 minutes.  Remove from the pot.  In the once-again empty pot, melt the remaining 2 Tbs. butter then whisk in the flour.  Cook, whisking constantly, for one minute.  Starting slowly, whisk in the half-and-half and the chicken broth, making sure to whisk so that there are no lumps; bring the mixture to a boil.  Reduce the heat to medium-low and simmer until the mixture thickens slightly, about 5 minutes.  Add the chicken and cook until the chicken is cooked through, about 10 minutes, or more, depending on the thickness of the chicken breasts.  Remove the pot from the heat, remove the chicken from the sauce and, once the chicken is cool enough to handle, shred it into bite-size pieces.  Add the cheeses to the sauce and whisk until smooth.  Add the shredded chicken, onions, noodles, peas, 1½ tsp. salt, and 1¼ tsp. pepper into the cheese sauce.  Pour the mixture into a 13 x 9-inch baking dish (the Pillivuyt lasagna baker is perfect), top with the crackers, and bake until the crumbs are browned and the mixture is bubbly, about 15 minutes.  Let the casserole cool for about 10 minutes, then serve.