Bring filet mignons to room temperature.
Preheat oven to 425°F.
Preheat cast-iron skillet over high heat. Season filet mignons with kosher salt and freshly ground black pepper on all sides.
Add avocado oil to cast-iron skillet and sear filet mignons on both sides, 3 minutes per side.
Add butter and rosemary to pan, and spoon herb butter over filet mignons for 30 seconds.
Remove filet mignons from cast-iron skillet and set aside, reserving 2 Tbsp of the pan juices.
Place carrot coins and smashed garlic into cast-iron skillet and simmer for 2 minutes.
Add 1 Tbsp sherry vinegar and place cast-iron skillet into oven for 20 minutes or until carrots are fairly tender.
Place filet mignons back in cast-iron skillet on top of carrots, and cook in oven for 10 more minutes until medium-rare, or meat thermometer inserted into thickest part of filet mignon reads 120°F.
Remove cast-iron skillet from oven. Set filet mignons aside to rest, place carrots in bowl and smash. Season with additional kosher salt and freshly ground black pepper as needed.
Place cast-iron skillet back on stove over medium heat. Add Brussels sprouts and peas to cast-iron skillet, then add remaining sherry vinegar and sauté for 3 minutes. Season with salt and pepper to taste.
Individually plate filet with vegetables or serve family style. Drizzle remaining brown butter on each plate or over the filets if serving family style.