Bring a small pot with rice and 11/2 cups water to a boil. Reduce to a simmer, cover, and cook until tender, 18-20 minutes. Transfer rice to prepared baking sheet, spread into an even layer, and refrigerate or freeze until cool enough to handle. While rice cooks, prepare ingredients.

Trim and thinly slice green onions on an angle, keeping white and green portions separate. Mince garlic. Remove any discolored outer leaves from bok choy and trim ends. Coarsely chop leaves. Quarter stem and cut into 1" pieces. Peel, trim, and cut carrot into very thin slices. Stem mushrooms and cut caps into 1/4" strips. Slice Fresno chile into thin rounds. Discard seeds if you prefer less spice.

Add cooled rice, panko, liquid egg, white portions of green onions, half the garlic (reserve remaining for vegetables), 1/2 tsp. salt, and 1/4 tsp. pepper to a mixing bowl. Mix until rice easily sticks together, smashing some rice as you go. Reserve prepared baking sheet; no need to change foil.

Wet hands before working with rice cakes to prevent sticking. Separate rice mixture into four balls. Press into 1/2" thick, 4" diameter patties. Place a large non-stick pan over medium-high heat. Add 2 Tbsp. olive oil and rice cakes to hot pan. Cook until browned, 3-5 minutes per side. Remove rice cakes to prepared baking sheet and roast until heated through, 6-8 minutes. Reserve pan; no need to wipe clean.

Return pan used to brown rice cakes to medium-high heat. Add 1 tsp. olive oil, bok choy stems, carrot, and mushrooms to hot pan. Stir often until lightly browned, 4-5 minutes. Add bok choy leaves, 1/4 cup water, remaining garlic, and Fresno chile (to taste). Stir occasionally until water evaporates and bok choy stems are tender, 2-4 minutes. Add teriyaki glaze, combine thoroughly, and remove from burner.

Place any remaining sauce on a plate. Place rice cakes on sauce. Place vegetables next to rice cakes, or on top. Garnish with green portions of green onions.