Preheat the grill. Mist a shallow rectangular baking dish with cooking spray. Bring a medium pot of salted water to a boil. Add the broccoli and cook until bright green and crisp tender, about 5 minutes. Drain well and run under cold water until cold. Drain well again and squeeze dry in paper towels.
Toss the broccoli in a bowl with the Gruyere and garlic. Insert a paring knife into the thickest part of each chicken breast to make a 3-inch deep pocket. Stuff each chicken breast with equal amounts of the broccoli mixture. Rub both sides of the chicken breasts with oil and thyme and season with salt and pepper.
Heat a large nonstick pan over medium-high heat until very hot, about 5 minutes. Put the chicken in the pan and cook until golden brown and just cooked through, about 6 minutes per side. If the chicken begins to brown too quickly, turn the heat down to medium to finish cooking through. Transfer to the baking dish.
Meanwhile, combine the milk and stock in a small pot, season with salt and pepper and bring to a simmer over medium heat. Stir the sherry and cornstarch until smooth and pour, whisking constantly, into the sauce. Cook until just thickened, about 2 minutes. Remove from heat and stir in the Parmesan. Top each chicken breast with 2 tablespoons of sauce and place under the broiler. Broil until lightly browned, 2 to 3 minutes.
Let the chicken rest for 5 minutes and then cut each breast in half on an angle. Whisk the mustard into the remaining sauce and spread a few tablespoons on each plate. Top with a halved chicken breast.