Step 1

Preheat broiler to high. Heat 1 tablespoon oil in a large skillet over medium-high. Season cutlets with ¾ teaspoon salt and the pepper. Working in batches, cook cutlets, flipping once, until golden brown, 3 to 4 minutes per side. Arrange cutlets in a 13-by-9-inch baking dish. Wipe skillet clean.

Step 2

Heat 1½ cups milk, thyme, and ¼ teaspoon salt in a saucepan over medium-high until bubbles form around edges. Whisk flour and remaining ½ cup milk in a bowl until smooth; add to saucepan. Cook, whisking often, until slightly thickened, about 3 minutes. Remove from heat; whisk in cheese until melted and smooth. Pour cheese mixture over cutlets; top with prosciutto and panko. Broil until golden brown and bubbly, 2 to 3 minutes.

Step 3

Meanwhile, add haricots verts, ¼ cup water, and remaining 1 tablespoon oil and ½ teaspoon salt to skillet. Cook over high until water evaporates and haricots verts are tender, 4 to 5 minutes. Serve baked chicken with haricots verts.