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Preheat oven to 400 degrees. Spray a roasting pan with cooking spray.
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Heat 1 tablespoon olive oil in a large skillet over medium heat. Add half of garlic and cook until fragrant, about 2 minutes. Add spinach and cook until wilted, stirring occasionally. Remove from heat and stir in 1 tablespoon tarragon. Transfer to a bowl to cool. (You can put in the freezer for a minute to expedite.)
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Toss asparagus with 1 tablespoon olive oil and salt and pepper. Set aside.
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Season chicken breasts with salt and pepper. Cut a slit into the side of each piece, being careful not to cut all the way through.
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When spinach mixture has cooled down, add ¾ cup shredded cheese and stir to combine. Stuff each chicken breast with ¼ of the spinach-cheese mixture.
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Heat remaining 2 tablespoons olive oil in the same skillet used for the spinach over medium-high heat. Add stuffed chicken and cook until browned on each side, 3-4 minutes per side. Turn off heat and transfer chicken to roasting pan.
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Bake for 10 minutes. Remove pan from oven and add the asparagus to the pan. Sprinkle remaining cheese over the top of chicken. Bake 10-15 minutes longer, depending on size of the pieces, until chicken reaches internal temperature of 165 degrees.
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While chicken is baking, make the pan sauce. In the same skillet used to brown the chicken, melt butter over medium heat. Add shallot and cook until soft, 3-4 minutes, stirring often. Add remaining garlic and cook until fragrant, about 2 minutes. Add wine and scrape up any brown bits from the bottom of the pan. Cook for 2 minutes. Add broth, cream, and remaining 1 tablespoon tarragon. Stir to combine. Cook, stirring occasionally, until sauce thickens, about 5 minutes. Taste and add salt and pepper as necessary.
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To serve: Divide chicken and asparagus between serving plates and spoon the pan sauce over chicken.