Bring a large pot of water to a rolling boil. Salt the water and have an ice bath ready (ice and water mixed in a large bowl). Add the green beans and boil for about 1 minute and 30 seconds, until bright green. Remove the green beans immediately from the pot with a slotted spoon and transfer to the ice bath. Let cool for about 30 seconds, and then immediately drain and set aside.
Melt 1/4 cup of unsalted butter in a nonstick skillet over medium heat. Add the garlic and and cook for about 1 minute or until fragrant and soft. Remove immediately from the heat and transfer to a large bowl. Add the beans to the bowl with garlic butter and toss to coat. Set aside.
In a medium saucepan, melt the remaining 2 tablespoons of unsalted butter. Whisk in the flour, and cook until golden, about 1 minute. Whisk in the whole milk, and allow to think for 1 minutes. Remove from heat. Stir in the cheddar cheese and stir until melted. Season with salt and black pepper and a pinch of paprika. Stir in the tarragon.
Transfer the green beans to a large platter. Pour the cheese sauce on top of the garlic butter green beans and garnish with chopped green onions. Enjoy immediately!