Pat the chicken breasts dry. Using a sharp knife, slice each chicken breast half into two horizontally so you have 4 cutlets. Sprinkle the chicken with salt and pepper.
Place a large skillet over medium-high heat and add 3 tablespoons oil. When the oil starts to shimmer, add the chicken, 1 garlic clove and 1 sprig thyme and cook until the chicken is golden and opaque, 2 to 3 minutes per side. Transfer everything to a plate, tent with foil to keep warm and set aside.
Return the skillet to medium heat and add the apricot preserves, broth, 1 tablespoon vinegar, 1 tablespoon mustard, 1 garlic clove and the remaining fresh thyme sprig. Season with salt and pepper. Whisk together the mixture and cook until the sauce is smooth. Reduce the heat to low as you prepare the white bean puree and arugula.
Make your bean puree. Put the cannellini beans, remaining garlic clove, lemon juice and lemon zest in a food processor. Add in 1/4 cup olive oil and pulse until smooth and creamy. Season with salt and pepper. If the bean puree isn't as creamy as you want, add 1 tablespoon water and pulse again. Set aside.
Chop the pistachios and set aside.
Whisk together the remaining 1 tablespoon vinegar and remaining 1 teaspoon mustard in a medium bowl. Whisk in the remaining 1/4 cup oil. Add the arugula and toss until coated with the dressing. Season with salt and pepper to taste and set aside.
Return the chicken and any juices on the plate to the skillet with the sauce. Increase the heat to medium. Cook until the chicken is cooked through, 2 to 3 minutes more, flipping the cutlets halfway through.
To serve, dollop and swoop some bean puree into a shallow bowl or deep plate. Slice a chicken cutlet and place it on top of the bean puree and top with the sauce. Place some arugula next to the chicken. Sprinkle everything with the chopped pistachios and drizzle with olive oil. Repeat with the remaining puree, chicken, arugula and pistachios. Enjoy!