Stir 1 Tbsp. chicken bouillon paste into ⅓ cup water in a small bowl until dissolved; set aside.

Heat 1 Tbsp. vegetable oil in a large high-sided skillet over medium-high heat. Cook 6 garlic cloves, thinly sliced, stirring, until fragrant, about 30 seconds. Add 2 medium heads of broccoli (about 2 lb.), florets cut into bite-size pieces, and cook, tossing constantly, until starting to turn bright green, about 30 seconds. Pour in reserved bouillon mixture and cover pan immediately. Cook until broccoli is crisp-tender, about 3 minutes.

Stir 2 tsp. cornstarch and 2 Tbsp. water in a small bowl. Add to pan along with 4 Tbsp. unsalted butter and 2 tsp. toasted sesame oil; toss to combine. Cook, uncovered, stirring often, until sauce is thickened and coats broccoli, about 2 minutes. Transfer to a large bowl.