Pat the chicken dry, season with salt and pepper, and heat 1 tablespoon butter plus the olive oil in a 12-inch skillet over medium-high heat; brown the chicken in a single layer 2–3 minutes per side until golden but not fully cooked, working in batches if needed, then transfer to a plate.

Reduce heat to medium, add remaining 1 tablespoon butter, then sauté the onion, carrots, and celery until softened and starting to caramelize, about 6–8 minutes; sprinkle the flour over the vegetables and stir for 1–2 minutes to cook the raw flour taste.

Slowly pour in the chicken broth while stirring to dissolve lumps and scrape browned bits from the pan, then add the milk and cream and return the chicken to the pot; bring to a gentle simmer and cook uncovered for 3–5 minutes to reduce slightly.

Add the gnocchi, gently stir to submerge, cover, and simmer over medium-low for 4–6 minutes until the gnocchi float and are tender; stir in frozen peas and thyme, adjust seasoning with salt and pepper, and add up to 1/4 cup extra broth if the sauce is too thick.

Serve hot from the pan with a sprinkle of parsley or grated cheddar if using; store leftovers in an airtight container in the refrigerator for up to 3 days.