Heat butter or olive oil in a large, deep pan or Dutch oven over medium-high heat.

Add the ground meat and cook, breaking it up with a spoon, until browned all over, about 7-10 minutes. Drain any excess fat.

Add the chopped onion and optional diced carrot to the pan. Cook until the onion softens, about 5-7 minutes.

Stir in the minced garlic and sliced mushrooms. Cook for another 5 minutes until mushrooms release their liquid and start to brown.

If using, pour in the wine and let it simmer for 2-3 minutes, scraping up any browned bits from the bottom of the pan. Let the alcohol evaporate.

Pour in the milk, salt, and pepper. Bring to a gentle simmer, then reduce heat to low.

Cover and simmer for at least 20-25 minutes, stirring occasionally, until the sauce has thickened slightly and flavors have melded. If using, stir in the heavy cream during the last 5 minutes of simmering.

Taste and adjust seasoning if needed.

Serve hot, garnished with fresh parsley or dill.