1) In a small to medium pot, add butter over medium heat. Once melted, add sliced shallot and cook for a minute. Stir in drained and rinsed beans, sun-dried tomatoes, and 1/4 cup water. Simmer for a few minutes and then mash lightly with a potato masher or fork. Some lumps are okay.
2) Toast bread and slice veggies thin.
3) For egg, add about 1/4 cup of olive oil in a large skillet. The olive oil should coat the entire pan. Heat over medium heat. Once hot, crack in eggs. Cook for about 2-3 minutes until eggs start to get crispy around edges and whites are mostly cooked. Then use a spoon to spoon the hot oil over each egg, cooking the top of the egg.
4) For serving, add a big smear of bean paste to each piece of toast. Top with sliced veggies and a drizzle of oil from sun-dried tomatoes. Add a few fresh tarragon leaves and a crispy fried egg on the side. Season with coarse sea salt and serve while warm.