Make mushroom-spinach mixture: Heat olive oil in a frying pan over medium heat. Add mushrooms. Sauté, stirring often, until well browned, 7 minutes.
Add 2 tsp. butter. When it foams and browns, add onion and garlic and cook until softened, 3 minutes. Add spinach and cook until it wilts evenly. Stir in vinegar and 1/4 tsp. salt, then spread on a plate and let cool. Put cooled mixture in a food processor and pulse to chop coarsely.
Make béchamel sauce: Heat milk over medium heat or in a microwave until just steaming. Meanwhile, put 1/3 cup flour and remaining 1/4 cup butter in a medium saucepan and heat over medium-low heat until butter is almost melted. Cook roux, whisking, until it is pale gold and smells toasty, 3 to 4 minutes. Slowly whisk hot milk into roux to create a very thick paste. Stir in nutmeg and 1 tsp. salt.
Remove from heat and fold in mushroom mixture. Season to taste with salt, scoop into a bowl, and press plastic wrap against surface to keep a skin from forming. Chill at least 2 hours and up to 2 days.
Form the croquetas: Scoop out 2-tsp. portions from bowl and roll into balls (they should be about 1 in. diameter). Chill until firmed up, at least 20 minutes.
Pour enough canola oil into a deep, wide pot so oil is 3 in. deep. Insert a deep-fry thermometer and heat oil to 350° over medium heat.
Meanwhile, coat croquetas: Set out 3 shallow bowls. Fill the first with remaining 1 cup flour, the second with egg wash, and the third with panko. Working with a few balls at a time, roll first in flour, then in egg wash (use a slotted spoon to roll them over and lift them out), and last in panko, pressing to help crumbs adhere (balls should be well coated). Transfer to a plate as done.
Using a metal slotted spoon, lower balls into oil. Fry in 2 batches until deep golden brown, about 5 minutes per batch, making sure oil temperature returns to 350° between batches. Drain on paper towels and sprinkle with salt.
Make ahead: Through step 5, up to 2 days, chilled; through step 6, up to 1 day, chilled; through step 8, up to 1 day, chilled.