Fill a large bowl with ice water. In a large, deep skillet, bring a few inches of water to a boil. Salt the water, add the vegetables and cook until crisp-tender, about 3 minutes. Drain; transfer to the ice water to cool, then drain again. Cut the asparagus and haricots verts on an angle into thirds.
Preheat a large cast-iron skillet over medium-high.
Pat the chicken dry. Season both sides of the chicken with salt and pepper.
Add 2 tbsp. olive oil to the skillet, two turns of the pan. Add the chicken, skin side down, and cook until browned and crispy, 4 to 5 minutes. Turn the chicken and cook until browned and cooked through, 4 to 5 minutes more; transfer to a plate. Reduce the heat to medium and add 2 tbsp. butter. When the butter is melted, add the sage and cook until crispy, about 2 minutes. Transfer the sage to a paper towel and let cool. Add the wine, garlic and mustard to the skillet and cook, swirling often, until fragrant, about 1 minute. Return the chicken to the pan and baste with the pan juices. Douse the chicken with 4 tbsp. lemon juice. Top with the sage. Remove from the heat.
Meanwhile, in a large skillet, heat the remaining 2 tbsp. olive oil, two turns of the pan, and the remaining 2 tbsp. butter over medium. Add the shallot and cook, stirring occasionally, until softened, 2 to 3 minutes. Add the blanched vegetables, season, then add the escarole and toss until wilted, about 3 minutes. Douse with the remaining 2 tbsp. lemon juice.
Serve the chicken with the vegetable medley alongside.