Heat olive oil in a medium pot over medium heat. Sauté onions and garlic until soft and fragrant.
Add diced potatoes and cook for 3 minutes, stirring occasionally.
Pour in vegetable broth, bring to a simmer, and cook until potatoes are tender, about 12 minutes.
Add chopped spinach and vegan feta; stir until spinach wilts.
Blend soup partially for creamy texture, then add plant milk. Season with salt and pepper.
For pudding, mix cornstarch with a splash of oat milk until smooth.
Heat remaining oat milk, coffee, and maple syrup in a saucepan over medium heat.
Whisk in cornstarch mixture and cook until thickened, about 5 minutes. Remove from heat; add vanilla and salt.
Chill pudding in the fridge while serving warm soup.