Place each chicken breast between two sheets of plastic wrap and gently pound with a meat mallet or rolling pin until approximately 1/2 inch thick. In a bowl, combine the finely chopped spinach, crumbled feta, grated Parmesan, Dijon mustard, minced garlic, salt, and freshly ground black pepper. Mix until evenly combined, making sure the spinach has been pressed to remove excess moisture.
Combine the softened unsalted butter, chopped parsley, thyme, lemon zest, salt, and pepper in a small bowl. Stir until evenly incorporated. Spread a thin, even layer of herb butter over each flattened chicken breast, leaving a small border around the edges.
Divide the spinach and feta filling evenly among the four chicken breasts. Arrange the filling in a horizontal line across the center of each piece. Starting from one short side, carefully roll the chicken around the filling. Secure each roulade with kitchen twine or toothpicks to maintain its shape during cooking.
Season the outside of the chicken roulades with salt and freshly ground black pepper. Heat 1 tablespoon of olive oil in a large skillet over medium-high heat. Add the secured roulades and sear for approximately 2 to 3 minutes per side, turning until the exterior develops a deep golden color.
Pour the optional dry white wine or chicken broth into the skillet and use a wooden spoon to loosen the browned bits from the bottom. Allow the liquid to reduce slightly. Add enough water to reach approximately halfway up the chicken roulades. Bring to a gentle simmer, reduce the heat to low, and cover. Cook for 8 to 10 minutes, or until the thickest part of the chicken reaches an internal temperature of 165°F (74°C).
Carefully transfer the cooked roulades to a warm plate and allow them to rest for 5 minutes. This helps the chicken retain its moisture and makes slicing easier. Remove all kitchen twine or toothpicks before serving.
Increase the heat beneath the skillet and allow the remaining cooking liquid to simmer for another 1 to 2 minutes until slightly thickened and concentrated in flavor. Taste the sauce and adjust the salt and pepper as needed. Spoon the finished pan sauce over the roulades.
Slice the rested chicken roulades diagonally to reveal the spinach and feta filling. Arrange the slices on serving plates with the spiral pattern visible. Serve alongside roasted baby potatoes, a simple green salad, or grilled asparagus. Garnish with optional lemon wedges and fresh parsley.