Step 1Trim the ends of the zucchini to make them a uniform length. Slice them lengthwise as thin as you can, about one-eighth inch (this is most easily done with a mandoline, but if you're careful, a very sharp knife will also work). You should have at least 24 thin strips of zucchini.
Step 2Place the zucchini in a bowl, salt generously and toss to coat, then transfer to a colander and set aside until the zucchini have softened, at least 30 minutes.
Step 3While the zucchini are sitting, whisk together 3 tablespoons olive oil, the minced garlic and lemon juice and season with a pinch of salt. Cut the cherry tomatoes in half and season lightly with salt.
Step 4Rinse the zucchini slices under cold running water, then pat dry with a paper towel. Return to the bowl and season with just enough of the olive oil-lemon mixture to moisten lightly.
Step 5Weaving the zucchini may sound complicated (as with weaving a lattice-top pie), but it is not difficult at all. You'll need 6 strips of squash for each plate. Arrange three strips of zucchini side-by-side on the first plate. Lift the middle strip and place one strip of zucchini perpendicular to the other strips and over the two outer strips, making an "H." Unfold the middle strip over the perpendicular strip. Fold back the two end pieces on one side and lay another perpendicular strip, then unfold the end pieces. Repeat at the other end, then use your fingers to gently push the pieces together to make a tightly woven mat of zucchini. Repeat for the three remaining plates.
Step 6Place the fresh goat cheese in a bowl and stir in the dried oregano and the remainder of the olive oil-lemon mixture to make a smooth, creamy mixture. If necessary, add a little more olive oil.
Step 7Divide the goat cheese mixture evenly among the four plates, spooning it in the center of the zucchini mat. Scatter the cherry tomato halves around the outside. Drizzle lightly with a little more good olive oil and sprinkle with coarse salt. Tear the basil leaves into small pieces and scatter over top. Serve at room temperature.