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Soak porcini in hot water (1 1/2 cups) in a small bowl until well softened, 20 to 30 minutes. Lift porcini out, squeezing liquid into bowl, then rinse porcini (to remove any grit) and coarsely chop. Strain soaking liquid through a sieve lined with a damp paper towel into a small saucepan and reserve.
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Preheat oven to 425°F with rack in middle.
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Cook shallots in 2 Tbsp oil with 1/4 tsp salt in a 12-inch nonstick skillet over medium heat, stirring occasionally, until golden, about 7 minutes. Add porcini, chopped thyme, and 1/4 tsp pepper and cook, stirring occasionally, 2 minutes. Add 3 Tbsp Marsala and boil until most of liquid has evaporated, then remove from heat.
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Add honey to soaking liquid in saucepan with thyme sprigs, remaining 2 Tbsp Marsala, and 1/4 tsp salt. Boil until syrupy and reduced to about 3 Tbsp, 15 to 20 minutes. Discard thyme sprigs.
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Meanwhile, remove excess fat from chicken and pat dry. Season inside and out with a rounded tsp salt and 3/4 tsp pepper. Stuff cavity with mushroom mixture and tie legs with string.
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Put chicken in a small (13- by 9-inch) roasting pan. Roast until an instant-read thermometer inserted into fleshy part of a thigh (do not touch bone) registers 170°F, about 1 hour. About 5 minutes before chicken is done, brush with all of honey glaze. If glaze begins to get too dark before chicken is cooked through, loosely cover with foil.
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While chicken roasts, peel potatoes and cut into 1/2-inch pieces. In cleaned nonstick skillet, heat remaining 3 1/3 Tbsp oil over medium heat, then add potatoes, stirring to coat. Stir in remaining 1/4 cup water and simmer, covered, stirring occasionally, 10 minutes. Remove lid and increase heat to medium-high. Add 1/2 tsp salt. Continue to cook, stirring occasionally, until liquid has evaporated and potatoes are golden brown all over, about 20 minutes more. Transfer to a baking sheet (do not cover).
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Transfer chicken to a cutting board to rest 15 minutes. Turn off oven and put potatoes in oven to keep warm.
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Meanwhile, straddle roasting pan across 2 burners and add flour. Cook over medium-high heat, stirring, 1 minute. Add broth and vinegar, stirring and scraping up brown bits. Boil mixture until reduced by about a third, about 4 minutes. Strain through a fine-mesh sieve into a gravy boat.
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Discard kitchen string from chicken. Scoop out mushroom mixture into a bowl, stir in potatoes, and serve with chicken.
