Carefully debone the chicken, starting with the backbone. Use a sharp boning knife to separate the meat from the bones, being careful not to pierce the skin.

In a large skillet, heat olive oil (2 tbsp) over medium heat. Add shallots (2 medium) and cook until softened, about 5 minutes.

Add the chopped mushrooms and thyme to the skillet. Cook until the mushrooms have released their moisture and are browned, about 10 minutes.

Remove the skillet from the heat and stir in the bread crumbs (½ cups), salt (1 tsp), and pepper. Let cool slightly.

Lay the deboned chicken skin-side down on a work surface. Season with salt (1 tsp) and pepper. Spread the mushroom mixture evenly over the chicken.

Carefully roll the chicken into a tight cylinder, tucking in the ends as you go. Tie the ballotine tightly with kitchen twine at 1-inch intervals.

Place the ballotine in a large pot and cover with chicken broth. Bring to a simmer and cook for 1 hour, or until the internal temperature reaches 165°F (74°C).

Remove the ballotine from the pot and let cool completely. Refrigerate for at least 2 hours before slicing.

Remove the twine and slice the ballotine into 1/2-inch thick slices. Serve chilled or at room temperature.