Preheat the oven to 350°F/180°C.
Place an oven-safe 12-in. fry or omelette pan in the oven.
Use a mandoline or box grater to shred the outer green leaves of the Brussels sprouts, discarding the hard white core. Pour water into a medium sauce pan and bring to boil.
Add the shredded Brussels sprout to the boiling water and blanch just until the sprouts turn bright green. Drain and set the sprouts aside.
Place the scored chestnuts into a medium bowl. Add the salt and 2 Tbsp. olive oil and toss to coat. Place chestnuts in the preheated fry pan and roast in the oven for about 15 minutes, or until the shell peels back where scored. Remove the pan from the oven and let it cool.
When the chestnuts are cool enough to handle, peel away the shells, crumble the nut meat into a bowl and reserve.
Place the cooled 12-in. omelette or fry pan on the range top at medium heat. When the rim of the pan is hot to the touch again, add the butter and 1 Tbsp. olive oil. Let the oil and butter heat for a moment.
Add the shallots and stir briefly. Add the Brussels sprouts and the chestnuts. Stir to heat all ingredients and serve.