Roast Brussels Sprouts and Squash: Preheat the oven to 425°F (220°C) with a rack in the center position. Place the quartered Brussels sprouts, cubed butternut squash, and herbs on a half sheet pan, drizzle with olive oil, salt, and pepper, and toss until evenly coated.
Roast for 20 to 25 minutes, tossing every 10 minutes or so, until they vegetables are evenly caramelized. Remove from the oven, discard the herb stems, and allow to cool slightly (if you haven’t finished cooking the grits or eggs by the time the vegetables are finished cooking, pop the roasted vegetables into the oven as needed to stay warm).
Prepare the Grits: As the vegetables are roasting, combine the whole milk, salt, and 1 tablespoon (15 g) butter in a large saucepan (roughly 4 quarts in size). Bring to a low simmer over medium-high heat. Once simmering, slowly pour in the grits with one hand, as you whisk continuously with the other.
Whisk continuously for until the grits have thickened considerably and they are cooked through. Break the goat cheese into pieces and add to the grits, along with the remaining tablespoon of butter, stirring until completely melted. Season the grits to taste with kosher salt and freshly ground black pepper.
Remove from heat and keep warm as you poach the eggs (if the grits get dry or too thick, add a splash of milk to loosen them - readjust seasoning accordingly).
Poach the Eggs: Fill a medium saucepan with cold water (the water should be at least 2-inches deep) and heat over medium heat until small bubbles begin to form at the bottom of the pan. This is the ideal temperature for poaching eggs. Don’t allow the water to reach a true simmer or boil (if it does, reduce the temperature). Set aside a large plate lined with a paper towel.
Break one egg at a time over a fine-meshed sieve set over a bowl. Allow any loose egg whites to fall away into the bowl – this will eliminate any loose whites from disintegrating in the cooking water and result in clean, tight poached eggs. Carefully transfer the egg into a small ramekin or cup (if using extremely fresh eggs, you can go directly to this step and skip the sieve step). Repeat with each egg (placing them into their own ramekin/cup).
Check and readjust the cooking water temperature as necessary. Gently tilt and slide the eggs from the ramekins into the cooking water. Ideally, you want to add all of the eggs to the cooking water fairly quickly, so that they cook consistently with one another. Set a cooking timer to three minutes. This time will produce a set white and a completely runny yolk. Using a slotted spoon, transfer the eggs from the water, one at a time, and set on the paper-towel lined plate to soak up any excess cooking water.
Spoon the grits into serving bowls, top with a large spoonful of roasted vegetables, and finally, top each plate with a poached egg. Garnish the place with fresh thyme leaves. Serve immediately.