Preparation Whisk mustard, Worcestershire sauce, garlic, 2 Tbsp. lemon juice, and 2 tsp. pepper in a medium bowl. Stream in 1/3 cup oil, whisking constantly until emulsified. Transfer half of marinade to a small bowl, then whisk in mayonnaise and cornichons; season dressing with salt. Set aside. Season chicken thighs all over with salt and place in bowl with remaining marinade; turn to coat. Let sit at room temperature, turning occasionally, at least 30 minutes, or cover and chill up to 4 hours. Prepare a grill for medium-high heat; lightly oil grate. Grill chicken, turning halfway through, until deeply browned and cooked through, 8–10 minutes. Transfer to a platter and let rest 5 minutes. Meanwhile, grill buns, cut side down, until lightly toasted, about 30 seconds. Transfer to platter with chicken. Toss sliced fennel, fennel fronds, basil, and remaining 2 tsp. oil and 2 tsp. lemon juice in a medium bowl to combine; season with salt and pepper. Spread reserved dressing over cut sides of each bun. Assemble sandwiches with buns, chicken, and fennel salad.
Preparation Whisk mustard, Worcestershire sauce, garlic, 2 Tbsp. lemon juice, and 2 tsp. pepper in a medium bowl. Stream in 1/3 cup oil, whisking constantly until emulsified. Transfer half of marinade to a small bowl, then whisk in mayonnaise and cornichons; season dressing with salt. Set aside. Season chicken thighs all over with salt and place in bowl with remaining marinade; turn to coat. Let sit at room temperature, turning occasionally, at least 30 minutes, or cover and chill up to 4 hours. Prepare a grill for medium-high heat; lightly oil grate. Grill chicken, turning halfway through, until deeply browned and cooked through, 8–10 minutes. Transfer to a platter and let rest 5 minutes. Meanwhile, grill buns, cut side down, until lightly toasted, about 30 seconds. Transfer to platter with chicken. Toss sliced fennel, fennel fronds, basil, and remaining 2 tsp. oil and 2 tsp. lemon juice in a medium bowl to combine; season with salt and pepper. Spread reserved dressing over cut sides of each bun. Assemble sandwiches with buns, chicken, and fennel salad.