Beer-Cheese Burgers

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For the cheese sauce:

2 tablespoons unsalted butter

2 tablespoons all-purpose flour

1/4 teaspoon mustard powder

3/4 cup beer

1/2 cup half-and-half

6 ounces grated sharp yellow cheddar cheese

1 tablespoon horseradish

For the burgers:

1 1/2 pounds ground beef, preferably chuck

Kosher salt and freshly ground pepper

Vegetable oil, for the grill

4 poppy seed rolls, split

Tomato slices, for topping

2 tablespoons chopped fresh chives