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Step 1
Pre-heat the oven to 400 degrees Fahrenheit. Heat a large pot over medium heat and add butter and the shallots. Cook for 2 minutes and then add the flour and stir until everything is combined. Next, add the broth, milk, salt and pepper. Cook mixture until thickened, stiring occasionally, about 10-15 minutes.
Step 2
Use a mandolin or sharp knife and slice the potatoes about 1/8 inch thick. Spread about 1/2 the sauce in the bottom of a 10 inch oven safe dish and top with a single layer of potatoes. Repeat steps until sauce is used up, ending with the sauce. Cover with foil and place into the oven. Cook potatoes until fork tender, 30-40 minutes.
Step 3
Make the crispy topping while potatoes are cooking. Melt the butter in a small saucepan and stir in the breadcrumbs, garlic, walnuts and salt. Cook for 2-3 minutes and remove from heat.
Step 4
Once potatoes are tender, take out of the oven and remove the foil. Add the basil into the topping mixture and evenly spread topping on the top of the potatoes. Place back in the oven, uncovered, for 5 minutes. Potatoes are best when served warm, right out of the oven.