Preheat oven to 375 degrees F. Butter the bottom and sides of a 10 by 10 by 2-inch baking dish with 1 tablespoon butter. Set aside.
Place the milk in a small saucepan and bring to a simmer over medium heat.
Melt the remaining butter in a medium saucepan over medium heat. Add the flour and whisk until combined with the butter. Let the flour mixture cook for about 60 seconds, do not allow it to get dark. Slowly whisk in the hot milk and continue whisking until the mixture begins to thicken. Let the sauce cook for at least 4 to 5 minutes, whisking constantly. Remove the sauce from the heat and whisk in the Monterey Jack, Cheddar and Parmesan cheeses until slightly smooth, season well with salt and pepper.
Place the cauliflower in a large bowl, add the cheese sauce, and poblano and stir until combined. Transfer mixture to the prepared pan and dot the top with the goat cheese. Bake until the cauliflower is tender and the top is lightly golden brown, about 30 to 40 minutes. Remove from the oven and let rest 10 minutes before serving.