To brine the chicken, in a bowl, mix together the water, salt, vanilla extract, and thyme. Place the chicken in a large gallon-size resealable plastic bag and add thee brine until nearly full. Press as much air out of the bag as possible and seal. Refrigerate overnight.
For the vanilla bean brown butter, melt the butter in a small saucepan over medium heat, stirring occasionally. Continue cooking the butter, stirring several times to ensure it cooks evenly, until it turns golden brown and smells toasty. Remove from the heat and use a blunt butter knife to scrape the inside of the vanilla bean pods over the pot, allowing the small vanilla beans to fall into the brown butter. After scraping, add the vanilla pods, vinegar, salt, and allspice to the pot and whisk. Set aside and infuse for 15 minutes.
To roast the chicken, preheat the oven to 425 degrees Fahrenheit. Evenly distribute the onion in a small roasting pan. Remove the chicken from the brine, rinse gently, and pat dry. Place the bird in the pan and rub with half of the brown butter mixture, inside and out, getting under the skin of the breast. Stuff with two of the pear quarters. Remove the whole vanilla bean pods from the pot and place inside the chicken as well. Truss the bird. Arrange the remaining pear quarters around the bird and drizzle the remaining brown butter mixture and stock over the pears and chicken.
Roast for 30 minutes. Reduce the heat to 375 and roast, turning halfway through, for 1 hour 15 minutes to 1 hour 45 minutes longer, until a thermometer inserted into a thigh of the bird reads at least 165 degrees Fahrenheit. If the bird browns too quickly, tent it with tin foil. Allow the chicken to rest for 20 minutes before carving.