Peel and slice shallots into very thin rounds. Trim and thinly slice green onion on an angle, keeping white and green portions separate. Peel and cut potato into 1" dice. Rinse steaks, pat dry, and season both sides with 1/4 tsp. salt and a pinch of pepper.

Place potatoes in a medium pot and cover with lightly salted water. Bring to a simmer and cook until fork tender, 15-18 minutes. Drain potatoes in colander and return to pot. Add 1/4 the evaporated milk (reserve remaining for sauce) and mash with a potato masher or fork until smooth. Add more evaporated milk, 1 Tbsp. at a time, to achieve desired consistency. While potatoes cook, caramelize shallots.

Heat 1 tsp. olive oil in a medium non-stick pan over medium heat. Add shallots to hot pan and cook, stirring often, until caramelized, 6-8 minutes. Stir caramelized shallots into mashed potatoes and season with 1/4 tsp. salt and a pinch of pepper. Reserve pan; no need to wipe clean.

Return pan used to caramelize shallots to medium-high heat. Add 1/2 tsp. olive oil and steaks to hot pan. Cook until well-browned and steaks reach a minimum internal temperature of 145 degrees, 3-5 minutes per side. Transfer to a plate and rest 5 minutes before serving. Wipe pan clean and reserve.

Return pan used to cook steaks to medium heat. Add 1/2 tsp. olive oil and white portions of green onion to hot pan. Cook until fragrant, 30 seconds. Add remaining evaporated milk and bring to a simmer. Cook, stirring often, until slightly thickened, 3-5 minutes. Tear smoked Gouda slices into large chunks and add to sauce. Stir constantly to melt. Remove from burner and season with 1/2 tsp. salt and a pinch of pepper.

Place a serving of mashed potatoes on a plate. Spoon sauce in front of potatoes and top with steak. Enjoy any remaining sauce on top of potatoes like a gravy. Garnish with green portions of green onion.