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1
Heat the oven to 400 degrees F and arrange a rack in the middle.
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2
Melt 1 tablespoon of the butter in a medium frying pan over medium heat.
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3
When it foams, add the shallots, season with salt and pepper, and cook, stirring occasionally, until softened and golden brown.
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4
Transfer to a plate and set aside.
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5
Add the port to the frying pan and bring to a boil over high heat.
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6
Reduce the heat to medium and continue to simmer until the port is reduced by half, about 6 to 10 minutes.
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7
Add the broth and reduce by half again (you should have about 2 cups); remove from heat and set aside.
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8
Meanwhile, mix together the reserved shallots, cheese, and 1 tablespoon of the thyme in a small bowl until evenly incorporated.
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9
Divide into 6 portions and set aside.
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10
Make pockets in the chicken by laying 1 breast on a cutting board and using a sharp knife to slice horizontally about three-quarters of the way through the meat.
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11
Move the knife in a fanning motion to slightly enlarge the pocket (be careful not to cut all the way through); repeat with the remaining chicken breasts.
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12
Season the chicken all over with salt and pepper and stuff the divided shallot filling into the pockets.
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13
Heat the olive oil and 2 tablespoons of the butter in a large frying pan over medium-high heat.
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14
When the butter foams, add 3 of the chicken breasts and cook until golden brown, about 3 minutes.
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15
Flip and repeat.
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16
Remove and repeat with the remaining chicken.
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17
Once all the chicken is browned, place it on a baking sheet and bake until its cooked all the way through (the juices will run clear when pierced with a knife), about 20 minutes.
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18
About 5 minutes before the chicken is ready, melt 1 tablespoon of the butter in a medium frying pan over medium heat.
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19
Once it foams, add the figs, cut side down, and the remaining thyme and cook until the figs are well browned.
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20
Add the reserved port sauce and the remaining 2 tablespoons butter and stir until the butter has melted and the sauce is heated through.
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21
Season with salt and pepper if necessary.
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22
When the chicken is ready, transfer to a serving platter, top with the sauce and figs, and serve.