Combine 2 cups warm water with the honey and yeast in a stand mixer fitted with the paddle attachment. Stir on very low speed until the yeast is dissolved. Increase the speed to low and let run until the yeast is bubbly. If the yeast is not foamy, it is not active and must be replaced. Turn off the mixer and allow the yeast mixture to sit for 2 minutes.
Add 1/2 cup oil and a couple large spoonfuls of the flour to the bowl. Mix on very low speed. Increase the speed to low and add the remaining flour, a couple of spoonfuls at a time. Continue to beat on low speed for 2 minutes. Scrape down the sides of the bowl. Increase the speed to medium-low and beat for another 3 minutes. Scrape down the sides of the bowl. Add the salt and remaining 2 tablespoons warm water and mix for 1 more minute.
Grease a large bowl with olive oil, just to coat. Add the dough to the bowl, then cover the bowl with a damp cloth or plastic wrap and allow to proof in a warm area until the dough is twice its size, 60 to 90 minutes.
Drizzle the remaining 1/4 cup olive oil onto a parchment paper-lined half-sheet pan. Transfer the dough to the pan. Stretch the dough into the corners, pressing down with your fingers to make dimples that don't pierce all the way through. Flip the dough over a couple of times to ensure an even coating of oil. Allow the dough to sit for 90 minutes for a second proof.
Preheat the oven to 450 degrees F.
Press your fingers into the dough once more to form dimples, then brush the focaccia with the melted butter. Arrange the vegetables and herbs to create a picture or pattern of your choice, from a garden to a sun—even a house. You can use the herbs as stems and the olives, tomatoes, peppers, onions and capers to form flowers. Lightly press the vegetables into the dough, being careful to not collapse the air. Sprinkle the top with a generous amount of sea salt. Bake for 20 minutes, then reduce the heat to 400 degrees F and continue to bake until the focaccia is golden, about 10 minutes longer.
Allow to cool in the pan for 30 minutes before serving.