Rinse the shrimp under cold water, drain in a colander, and pat dry with paper towels; set aside.
In a small bowl, whisk together 4 tablespoons San-J hoisin sauce, 1/4 cup water, and 1 teaspoon cornstarch until smooth; set the sauce mixture aside.
Heat a large skillet or wok over medium heat until hot. Add 1 1/2 tablespoons cooking oil and swirl to coat the pan.
Add the 6 slices of peeled ginger to the pan and sauté about 30 seconds, until fragrant.
Add the 12 oz (340 g) shrimp to the pan. Stir and toss continuously until the shrimp turn opaque and the surface turns white (about 1–2 minutes).
Add the 6 oz (170 g) broccoli florets and the 1 small red bell pepper (cut into pieces). Stir to combine and cook 1–2 minutes, until the vegetables are bright and slightly tender.
Pour the hoisin sauce mixture into the skillet. Stir to coat all ingredients and simmer, stirring occasionally, until the sauce thickens and becomes glossy (about 1–2 minutes) and the shrimp are fully cooked.
Turn off the heat, transfer the hoisin shrimp and broccoli to a serving platter, and garnish with white sesame. Serve immediately.