Line a large plate with paper towels. In a small saucepan, heat the oil over medium-high heat until it begins to shimmer and lightly smoke. Add the shallots and cook, stirring often, until light golden brown, about 7-9 minutes. Using a slotted spoon, transfer the shallots to the paper towel-lined plate. Sprinkle with ¼ teaspoon salt.

Preheat the oven to 400°F. Butter a 2- to 3-quart casserole dish.

Bring a large pot of salted water to a boil. Add the green beans and cook until crisp-tender, 4-5 minutes. While the beans cook, fill a large bowl with ice water. Drain the beans and immediately plunge them into the ice water to stop cooking. Transfer to a clean kitchen towel to dry.

In a large skillet, melt the butter over medium-high heat. Add the mushrooms and toss. Cook, stirring occasionally, until golden brown, about 5 minutes. Add 1 teaspoon salt, ½ teaspoon pepper and the minced garlic; cook 1 minute. Sprinkle in the flour and cook, stirring, 30 seconds. Whisk in the broth and milk and bring to a simmer. Cook 5-6 minutes, or until thickened. Turn off the heat and add half of the shredded Comté, along with ½ teaspoon salt and ¼ teaspoon pepper; stir until the cheese is melted. Add the green beans and stir to coat.

Transfer the mixture to the prepared baking dish. Sprinkle the remaining Comté over the top. Bake 10 minutes. Before serving, sprinkle the fried shallots over the gratin. Serve warm.