Warm the milk with the bay leaf in a small saucepan to just under a simmer.

In another saucepan, melt the butter over medium heat. Add the flour to the butter and whisk constantly to combine without browning the flour.

Remove the bay leaf from the milk, then gradually add the milk to the butter and flour mixture while whisking until a smooth consistency is reached and there are no lumps. Cook over medium-low heat for 10-15 minutes, whisking or stirring often.

Finally, add the salt, nutmeg, and pepper and stir to fully incorporate. Taste and adjust salt as necessary.

In a large skillet over medium heat, melt the butter. Add the vegetables and salt and gently cook until the vegetables soften but are not mushy – about 5 minutes. Stir the vegetables into the béchamel sauce with the Pecorino and Parmigiano cheeses until thoroughly incorporated.

Preheat your oven to 375°F.

Assemble your lasagna starting with a thin layer of olive oil on the bottom and sides of a 9x13-inch baking dish. Add a layer of pasta, then sauce/vegetable mixture. Repeat until your dish is full; the top being a thin layer of sauce.

Sprinkle the top with more Parmigiano or Pecorino (or both) if desired. Place in the oven for 30-45 minutes or until the lasagna is bubbly and the center is hot (an instant read thermometer should read 170°F in the center).

Once done, allow your lasagna to sit for about 10-15 minutes before cutting into to allow it to cool slightly.