Put 3/4 cup oil in a blender, add a few basil leaves, and pulse to blend. Continue pulsing as you add more leaves and 1 small ice cube (it helps move the leaves around); blend until mixture is smooth. Season with about 1 tsp. salt.
Put cucumbers and lettuce in a large bowl, add 5 to 6 tbsp. oil and the lemon juice, and toss to coat evenly. Season with about 1/4 tsp. salt and toss again.
Spread basil purée on plates or a large platter, top with greens and cucumbers, and crumble goat cheese on top. Serve right away.
*Find Little Gem lettuce, a mini romaine, at farmers' markets.
Make ahead: For basil purée and lettuce, up to 1 day, chilled; bring basil purée to room temperature.