Preheat the broiler.
Heat a large nonstick skillet over medium-high heat. Add 1 1/2 tablespoons of the oil and the mushrooms and sauté for about 6 minutes, or until the mushrooms are golden and tender. Add the shallots, garlic and thyme, and sauté for about 3 minutes, or until the shallots begins to soften. Add the wine and simmer for about 20 seconds, or until it evaporates. Sprinkle the flour over the mushrooms and stir to blend. Stir in the milk and cream and bring to a boil, then reduce the heat to medium and simmer gently, stirring, for about 3 minutes, or until the sauce thickens slightly. Stir in 1/4 cup of the cheese and remove from the heat. Season to taste with salt and pepper.
Toss the broccolini into the hot mushroom mixture. Transfer the mixture to a 9-inch square baking or oval gratin dish.
In a medium bowl, mix the panko, remaining 3/4 cup cheese, and remaining 1 tablespoon oil, tossing well, then sprinkle the mixture over the top of the casserole. Broil, uncovered, watching closely, for 2 to 3 minutes, or until the bread crumb mixture is golden brown.
Make-Ahead: The panko mixture can be made up to 2 days ahead, covered and refrigerated. The broccolini can be blanched up to 1 day ahead, drained, covered, and refrigerated.
*Blanching Basics: Don’t be afraid of fancy culinary terms like “blanching.” Basically all you’re doing is cooking food, like broccolini, in boiling salted water for just a few minutes, or until its color brightens, then plunging it into ice water to stop the cooking. It’s necessary in this recipe to form the delicious creamy sauce—if the broccolini is not blanched first, the sauce will curdle—so don’t skip it.