Smash the cucumbers using a tortilla press, mallet, or your hands until they break into rough bite sized pieces. Transfer to a bowl.
Toss the cucumbers with kosher salt and let sit for 10 minutes to draw out excess moisture. Drain off the liquid and lightly pat dry if needed.
Add ume vinegar, scallion tops, umeboshi paste, and black sesame seeds to the cucumbers. Toss well to combine and adjust seasoning if needed.
Toss the sliced spring onion bottoms in potato starch until evenly coated, then shake off any excess starch.
Place the coated onions in a pot and cover with neutral oil. Slowly bring the oil up to temperature over medium heat, stirring occasionally, until the onions turn golden brown and crispy.
Immediately strain the onions from the oil and season with salt and black pepper. Reserve the hot onion oil for the chili oil.
Pour the hot onion oil over the gochugaru and sambal in a heatproof bowl. Let it cool slightly, then stir in the garlic chips.
To assemble, plate the dressed cucumbers, top with crispy onions, and spoon the chili oil over the top. Serve immediately.