In a medium saucepan over medium-high heat, bring water to a boil. Add the beans and partially cook for 4 to 5 minutes (depending on the degree of doneness you want). Remove from heat. Drain the beans in a colander and immediately immerse them in an ice water bath to stop from cooking; drain and set aside.
In a small saucepan over medium heat, combine 1 1/2 teaspoons of the olive oil and vinegar, gorgonzola, brown sugar, garlic, green onion, thyme, and basil. Cook just until the ingredients start to combine, approximately 7 minutes. Remove from heat.
Toss the green beans in the remaining 1 1/2 teaspoons olive oil and season lightly with salt and pepper. Lightly grill the beans on the barbecue grill approximately 1 to 2 minutes, stirring constantly. Remove from barbecue and toss in the warm vinaigrette. Transfer to a serving platter and serve immediately.