Heat oven to 425°F. Toss mushrooms with their tablespoon oil and 1/4 teaspoon salt on a sheet pan and roast for 18 to 22 minutes, turning once, until browned and no loose liquid remains.
Bring poaching water and the remaining 1/4 teaspoon salt to a gentle simmer. Add chicken, partly cover and cook for 15 to 20 minutes until the thickest centers reach 165°F; drain, rest for 5 minutes and shred finely with two forks.
Toast walnuts in a dry skillet over medium-low heat for 3 minutes and cool. Pulse walnuts, arugula, basil, garlic, Parmesan and lemon juice in a food processor, then pulse in the pesto olive oil to make a thick, slightly coarse sauce.
Fold the pesto through the warm shredded chicken. Reduce oven to 350°F and place croissant bottoms on a sheet pan.
Divide chicken over the bottoms, add roasted mushrooms and cover with Fontina. Set the tops beside them and bake for 5 to 7 minutes until cheese softens, removing the plain tops early if they color.
Close the sandwiches with the warmed tops and serve immediately.