Rinse the shrimp under cold water, drain well, and pat dry with paper towels; set aside.
In a small bowl, add the 1 teaspoon cornstarch to the 1/4 cup water and stir to dissolve. Add the 4 tablespoons San-J hoisin sauce and whisk until smooth; set the sauce mixture aside.
Heat a skillet or wok over medium heat. Once hot, add the 1 1/2 tablespoons cooking oil.
Add the 6 slices of peeled ginger to the oil and sauté until aromatic, about 30–60 seconds.
Add the shrimp to the skillet. Stir and toss constantly until the shrimp turn pink and opaque and the surface is no longer translucent (about 2–3 minutes).
Add the 6 oz broccoli florets and the 1 small red bell pepper (cut into pieces). Stir to combine and cook until the vegetables are tender-crisp, about 2–3 minutes.
Pour the hoisin–cornstarch sauce into the skillet and stir to coat all ingredients. Bring to a gentle simmer and cook, stirring occasionally, until the sauce thickens and becomes glossy, about 1–2 minutes.
Remove the skillet from the heat, transfer the shrimp and vegetables to a serving platter, and garnish with white sesame seeds. Serve immediately.