Heat the oven to 400°F and arrange a rack in the middle.
Melt 1 tablespoon of the butter in a medium frying pan over medium heat. When it foams, add shallots, season with salt and freshly ground black pepper, and cook, stirring occasionally, until softened and golden brown. Transfer to a plate and set aside.
Add port to the frying pan and bring to a boil over high heat. Reduce heat to medium and continue to simmer until port is reduced by half, about 6 to 10 minutes. Add broth and reduce by half again (you should have about 2 cups); remove from heat and set aside. Meanwhile, mix together reserved shallots, cheese, and 1 tablespoon of the thyme in a small bowl until evenly incorporated. Divide into 6 portions and set aside.
Make pockets in the chicken by laying 1 breast on a cutting board and using a sharp knife to slice horizontally about 3/4 of the way through the meat. Move the knife in a fanning motion to slightly enlarge the pocket (be careful not to cut all the way through) and repeat with remaining chicken breast. Season chicken all over with salt and pepper and stuff divided shallot filling into pockets.
Heat olive oil and 2 tablespoons of the butter in a large frying pan over medium-high heat. When butter foams, add 3 chicken breasts and cook until golden brown, about 3 minutes. Flip and repeat. Remove and repeat with remaining chicken. Once all chicken is browned, place on a baking sheet and bake until chicken is cooked all the way through (the juices will run clear when pierced with a knife), about 20 minutes.
About 5 minutes before the chicken is ready melt 1 tablespoon of the butter in a medium frying pan over medium heat. Once it foams, add figs, cut side down, and remaining thyme and cook until figs are well browned. Add reserved port sauce and remaining 2 tablespoons butter and stir until butter is melted and sauce is heated through. Season with salt and pepper if necessary. When chicken is ready, transfer to a serving platter, top with sauce and figs, and serve.
Beverage pairing: Qupé Marsanne, California. Marsanne makes wines with good weight, but also good acidity and subtle, unprepossessing flavors of white peaches and jasmine. It’s lovely with something equally humble as chicken, especially when accented with thyme, the sharpness of chèvre, and the richness of figs.
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