Preheat your oven to 350°F / 175°C. Lightly grease a 3-quart (9x13 inch) rectangular baking dish.
Cook the lasagna noodles according to package directions until al dente.
Drain them in a colander, rinse with cold water to stop the cooking, and drain again. Lay the noodles in a single layer on a sheet of foil to prevent them from sticking together.
While the pasta cooks, prepare the sauce.
In a medium saucepan, create a smooth slurry by whisking together 1 cup / 240 ml of the milk and the flour.
Once smooth, stir in the remaining 2½ cups / 600 ml (20.3 fl oz) milk, salt, dry mustard, pepper, and hot pepper sauce.
Cook and stir constantly over medium heat until the sauce thickens and begins to bubble.
Remove from the heat and stir in the Parmesan cheese until melted and smooth.
Now it's time to assemble!
Spread a thin layer of sauce on the bottom of your prepared baking dish.
Arrange one-third of the cooked lasagna noodles (3 noodles) over the sauce.
Top with half of the broccoli, half of the carrots, ¾ cup of the cheddar cheese, and about 1 cup (240 ml) of the sauce.
Repeat the layers once more: noodles, the remaining broccoli and carrots, another ¾ cup of cheddar, and another cup of sauce.
Finish with the last layer of noodles, the remaining sauce, and the final ½ cup of cheddar cheese sprinkled evenly over the top.
Bake, uncovered, for 35 to 40 minutes, or until the sauce is bubbly and the top is golden brown.
Let the lasagna stand for 15 minutes before slicing and serving. This allows it to set, making it much easier to cut into clean portions.
Squeeze the thawed broccoli thoroughly to remove as much water as possible. This is the key to preventing a watery lasagna and ensures a rich, concentrated flavor.
