Preheat the oven to 425 degrees, and line a 9 by 13 inch baking dish with parchment paper.
Grate the zucchini by hand or with a food processor. Place it in a fine mesh sieve over a bowl and line it with a thin kitchen towel. Put the zucchini in the strainer and let it sit for a few minutes. Then wrap the towel with the zucchini into a ball, and squeeze to absorb some of the excess water. Place the wrapped ball of zucchini in the strainer, and let it continue to drain while you prepare the other ingredients.
In a large bowl, whisk the eggs, then stir in the flour, baking powder, and salt. Mix to combine well. Then stir in the strained zucchini and the chives.
Spread the zucchini mixture evenly in the prepared baking pan on the parchment paper. The batter will be about 1/2 inch to 3/4 inch thick.
Bake at 425 degrees for about 25 minutes, until it’s set on the surface and it’s beginning to turn brown at the edges.
Leave the oven set at 425 degrees, and remove the pan from the oven. Lift the parchment paper to remove the zucchini “bread” to a cutting board. Cut the zucchini mixture into 8 pieces. (Cut in half the long way, then cut the top and bottom sections in quarters.)
Cut an 8 ounce block of cheddar in half. Then cut each half into 4 slices. Place 4 pieces of the zucchini bread back in the pan, and top them with the cheese (2 slices of cheese on each zucchini base). Place the remaining 4 zucchini sections on top of each sandwich, and return them to the oven. Cook for 5 more minutes, or until the cheese is melted. Serve warm.