Preheat the oven to 425°F. Halve the sprouts and let them rest for 20–30 minutes if you have the time. Pat them completely dry so they roast instead of steam.
Spray the baking sheet with olive oil then lay the Brussels sprouts cut side down in a single layer, leaving a little space between them. Season with the salt and pepper.
Roast for 20–25 minutes without stirring for the first 15, until the cut sides are deep golden brown and the outer leaves are crisp.
Meanwhile, melt the butter in a small saucepan over medium heat. Swirl as it foams for 3–4 minutes, until the milk solids turn golden brown and it smells toasty and nutty.
Take the pan off the heat and carefully add the balsamic vinegar (it will sputter). Swirl for 30 seconds until it comes together into a glossy, slightly thickened sauce.
Transfer the sprouts and walnuts to a serving dish, pour the balsamic brown butter over the top and toss gently. Finish with a pinch of flaky salt and the chopped walnuts and serve warm.