Bring a large pot of water to a boil for the pasta.
Heat a large nonstick skillet over medium-high. Add the oil, two turns of the pan. Add the chicken and season with salt and pepper. Cook, stirring occasionally, until lightly browned, about 3 minutes per side; transfer to a plate.
Add the butter to the skillet. When it melts, add the mushrooms and cook, stirring often, until lightly browned, about 2 minutes; season. Add the shallots and garlic. Cook, stirring often, until the shallots soften, 2 to 3 minutes. Sprinkle with the flour and stir to combine. Whisk in the sherry and cook, stirring often, until reduced by half, about 1 minute. Add the milk and stock and boil, stirring occasionally, until the sauce thickens, about 8 minutes. Add the dry mustard and a little sprinkle of nutmeg to the sauce; stir in the cheese and season. Return the chicken to the skillet; reduce the heat to low.
Salt the boiling water, add the pasta and cook to almost al dente. Reserve 1/2 cup of the starchy cooking water, then drain the pasta.
Add the peas and tarragon to the sauce and return to a bubble. Toss in the pasta, adding some of the reserved starchy cooking water if too thick. Serve in shallow bowls.