Melt 1 tablespoon butter in a medium saucepan over medium-high heat. Add the onion and garlic and cook, stirring occasionally, until softened, about 5 minutes. Add the rice and stir until lightly toasted, about 1 minute. Add the chicken broth and bring to a full boil. Cover, reduce the heat to low, and cook for 18 minutes.
Meanwhile, melt the remaining tablespoon butter in a small saucepan over medium heat. Add the flour and stir until incorporated. Gradually whisk in the milk and bring to a simmer. Simmer until slightly thickened, about 2 minutes. Remove from the heat and whisk in the Gruyere; keep warm, covered.
When the rice is cooked, stir in the cheese sauce, parsley, chives and tarragon. Season with salt and pepper.