Place the shrimp in a colander and rinse with cold water. Line a large plate or tray with paper towels and place the shrimp on the towels. Pat dry with additional paper towels. Set aside next to the stove.
Place the broccoli in a glass bowl with about 2 tablespoons of water. Loosely cover the bowl with a lid or plate. Microwave on high for 3-4 minutes, or just until the broccoli is tender. When the broccoli is bright green and barely fork tender, remove it from the microwave and set aside.
In a small bowl or measuring cup, whisk together the honey, soy sauce, and cornstarch. Add the sesame oil, sriracha (if desired), and pepper. Set aside.
Melt 2 tablespoons of butter in a large skillet over medium-high heat. Add the garlic and ginger and cook until fragrant, about 30 seconds. Add the shrimp to the pan. Cook without touching for about 2 minutes. Stir the shrimp and cook an additional minute, just until opaque and cooked through. Transfer the shrimp to a plate.
Add the sauce to the empty pan and increase the heat to high. Cook 2-3 minutes, just until thickened and a spatula leaves a trail when drawn across the bottom of the pan. Remove from the heat, add the shrimp and broccoli and toss to coat. Serve over rice.