In a large bowl, combine all of the ingredients for the crab cakes, except the butter. Divide the mixture into 4 large crab cakes or 6 small crab cakes. (At this point, you can place the made crab cakes in the refrigerator until you are ready to cook them.)
Melt the butter in a large skillet over medium-high heat. Cook the crab cakes in the butter for 2-3 minutes on each side, being careful when you turn them. Remove from the pan and drain the extra oil on paper towels.
Add the oil to a small saucepan over medium-high heat. Once the oil is sizzling, add the tomatoes and cook turning frequently until the tomatoes become blistered and start to burst. Season with the salt. Remove from the heat and place on top of the crab cakes.
Bring a medium pot of water to boil. Once the water is boiling, reduce the heat to a simmer. Crack each egg separately and add them one at a time to the simmering water. Cook for 3-5 minutes and remove with a slotted spoon. Place on top of the crab cakes and tomatoes.
In a small saucepan over medium heat, place the white vinegar, white wine, shallots, peppercorns, and two tarragon sprigs. Cook for 5-10 minutes until the liquid reduces to about 1/4 cup. Strain the mixture and place into a small liquid measuring cup that is no larger around than an immersion blender. Add the eggs to the mixture.
Place the butter in another bowl and melt. Once the butter is melted, use your immersion blender in the egg/liquid mixture and start to blend, slowly adding the butter while blending. The mixture will start to emulsify and thicken. Once the mixture is thickened, add the diced tarragon leaves. Season with salt and pepper. Serve over the crab cakes.