Trim bottom end from broccolini. Starting halfway down stalk, cut lengthwise through bottom end. Peel potato and grate on large hole side of a grater onto towel. Gather corners of towel together, lift, and spin base of towel to wring out as much liquid as possible. Trim and thinly slice green onions on an angle, keeping white and green portions separate.
Cook the Rice
Toss broccolini with 1 tsp. olive oil and a pinch of salt and pepper in a mixing bowl. Using a wire-mesh strainer, rinse rice until water is clear. Bring a small pot with 1½ cups water, rice, and a pinch of salt to a boil. Once boiling, add broccolini on rice, reduce to simmer, cover, and cook until tender and water has been absorbed, 18-20 minutes. Remove lid and set aside. Reserve bowl; no need to wipe clean. While rice cooks, make fritter mixture.
Make the Fritter Mixture
Add potatoes, carrots, liquid egg, green portions of green onions, 1½ Tbsp. rice flour (reserve remaining for gravy), furikake, ½ tsp. salt, and a pinch of pepper to mixing bowl used for broccolini. Combine thoroughly.
Cook the Fritters
Place a large non-stick pan over medium-high heat. Add 1 Tbsp. olive oil to hot pan. Separate fritter mixture into four balls, form into 4" patties, and carefully place into hot pan. Cook undisturbed until browned and crispy, 3-4 minutes per side. Remove to a plate. Reserve pan; no need to wipe clean.
Make the Gravy
Return pan used to cook fritters to medium-high heat. Add sesame oil and white portions of green onions to hot pan. Cook until fragrant, 30-60 seconds. Stir in remaining rice flour. Add soy sauce and 1 cup hot water to pan. Bring to a boil and cook until sauce thickens just enough to coat back of a spoon, 1-2 minutes.
Finish the Dish
Place rice and broccolini on a plate. Shingle fritters next to rice. Top with gravy.