How to Make It
Place the shiitake caps in a small heatproof bowl. In a saucepan, bring the vinegar, salt, sugar and 1/2 cup of water to a simmer, stirring to dissolve the sugar. Pour the brine over the mushrooms; let stand for 15 minutes. Drain and squeeze dry. Discard the brine.
In a blender, puree the dill with the oil until smooth. Strain through a fine sieve into a small bowl and season with salt.
In a small saucepan, cook the anchovies, garlic and lemon juice over low heat, stirring, until the anchovies dissolve, 5 minutes. Whisk in the butter a few cubes at a time. Stir in the dill. Cover; keep warm.
Light a grill. In a bowl, toss the summer squash, cucumber and shallots with the canola oil and season with salt and pepper. Grill over moderately high heat, turning, until the vegetables are lightly charred and crisp-tender: 5 minutes for the cucumber and shallots, 8 minutes for the squash. Transfer the vegetables to plates; add the mushrooms.
Meanwhile, in a saucepan of salted boiling water, blanch the green beans until crisp-tender, about 3 minutes. Drain and cool under cold water; pat dry.
In a large skillet, melt the butter. Add the green beans and radishes and cook over moderate heat until golden brown, about 3 minutes. Stir in the tomatoes and cook for 1 minute. Season with salt and pepper. Transfer to the plates with the vegetables. Drizzle some of the dill oil on top and garnish with the herbs. Serve with the bagna cauda, the remaining dill oil and bread.