• Preheat your oven to 375-F and butter the insides of a shallow baking dish large enough to hold the carrots stacked 2-3 high.
• Bring a large pot of salted water to a boil and parboil the peeled and trimmed carrots for 3-4 minutes or until they are still quite firm but nearly tender-crisp.
• Drain the carrots, cool them under cold running water, and then transfer them to the prepared baking dish.

To make the sauce

• Make the sauce by melting butter in a saucepan and saute the shallots until they're softened; 3-4 minutess. Add the flour, cook for another minute, and then whisk in the chicken or vegetable broth. Return to a simmer until thickened. Whisk in the lemon juice and Dijon mustard, strain the sauce and discard the shallots.
• Season the sauce with salt and pepper and pour the sauce over the carrots.

To make the crumb topping

• Break the sliced bread apart with your fingers into large crumbs.
• Melt the butter and stir in the remaining Dijon mustard, horseradish, shredded Parmesan cheese, thyme, salt and pepper.
• Toss the butter mixture with the crumbled bread, then sprinkle the bread topping over the carrots.
• Bake the Carrots au Gratin at 375-F for about 15-20 minutes or until it is hot and bubbling and the topping is golden brown.
• Serve and enjoy!