Season your chicken pieces generously with salt and pepper. Trust me, seasoning at this stage makes all the difference!
Heat 1 tablespoon each of olive oil and butter in a large skillet over medium-high heat until the butter is melted and slightly bubbly.
Add the chicken pieces in a single layer (you might need to work in batches). Cook for about 4-5 minutes, turning occasionally, until golden brown on all sides and just cooked through. Transfer to a plate and set aside.
In the same pan, add the remaining olive oil and butter. Once melted, add the onions and cook for 2-3 minutes until they start to soften.
Add the bell peppers and mushrooms, cooking for another 5-6 minutes until the vegetables soften and the mushrooms release their moisture.
Stir in the garlic and cook for just 30 seconds until fragrant - be careful not to let it burn!
Sprinkle the flour over the vegetables and stir constantly for about 1 minute. This creates the base for our sauce, and I find that cooking the flour a bit removes that raw taste.
Slowly pour in the chicken broth while whisking continuously to prevent lumps. Bring to a gentle simmer and cook for 2-3 minutes until it starts to thicken.
Reduce heat to low and stir in the half-and-half. Simmer gently for about 5 minutes until the sauce thickens enough to coat the back of a spoon.
Return the chicken to the pan along with any accumulated juices (that's flavor gold!).
Add the frozen peas, lemon juice, herbs, and nutmeg. Stir gently to combine and warm through for 2-3 minutes.
Taste and adjust seasonings with salt and pepper. The sauce should be creamy, savory, and just bright enough from the lemon.