Preheat oven to 400°F. Spray a baking sheet with 1-inch sides, with cooking spray or rub with olive oil.
Toss mushrooms and shallots with olive oil and season with salt and pepper. Turn out onto prepared baking sheet. Roast for 25 minutes, or until the mushrooms and shallots begin to brown on the edges. The shallots may take a little bit longer, so just remove the mushrooms and place on a plate until the shallots are roasted.
Remove both mushrooms and shallots from roasting pan and set aside to cool. Pour 1 cup of boiling water over the entire pan and scrape up any brown bits to use in your sauce. Place the liquid and brown bits into a frying pan along with 1/4 of the roasted shallots.
When veggies are cool, place them in the bowl of your food processor and pulse until chopped but still a little bit chunky.
Add parsley, garlic and spinach or arugula and pulse again to make the stuffing mixture. Add Boursin and pulse just until integrated.
Preheat the oven to 350°F or the barbecue to medium-low to medium, if baking on the grill.
Lay chicken breasts out flat on a cutting board, with the skin side down. Spoon the stuffing in the middle right between the breasts from top to bottom.
Cut 12 pieces of kitchen string, long enough to wrap around both sides of the chicken breast. Now, using one string at a time, roll the two sides of the split breast together and tie around in three or four places, making sure to keep as much of the stuffing inside the chicken roulade.
Then, bring the ends toward the center and tie two strings lengthwise around the rolled chicken breasts. Now you should have nice, neatly rolled double chicken breasts with the stuffing tucked inside.
Place stuffed breasts in a pan and bake at 350°F for 40-45 minutes in the oven until a meat thermometer reaches 165°F when inserted into the thickest part of the chicken. Then turn the oven to broil and cook just another minute or two under the broiler to give it some good color.
If you are grilling the chicken indirectly on the barbecue, heat the two side burners to high, place the stuffed chicken breasts in an aluminum pan and place the pan over the middle burner, which is turned off. Grill for 35 minutes, or until a meat thermometer reaches 165°F when inserted into the thickest part of the chicken. Then, take the roulades out of the pan and cook directly on one of the high burners for 1 minute per side for color.
To serve, slice the stuffed breasts into 1 1/2-inch slices and drizzle sauce over the top.