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.
Preheat oven to 425°F with rack in middle.
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.
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.
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.
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.
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).
Transfer chicken to a cutting board to rest 15 minutes. Turn off oven and put potatoes in oven to keep warm.
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.
Discard kitchen string from chicken. Scoop out mushroom mixture into a bowl, stir in potatoes, and serve with chicken.