Brush large heavy-duty rimmed baking sheet with olive oil.
Cut off all but 1/2 inch of green radish tops; reserve trimmed tops and rinse them well, checking for grit.
Coarsely chop radish tops and set aside. Cut radishes lengthwise in half and place in medium bowl. Add 1 1/2 tablespoons olive oil and toss thoroughly to coat.
Place radishes, cut side down, on prepared baking sheet; sprinkle lightly with coarse salt. Roast until radishes are crisp-tender, stirring occasionally, about 18 minutes.
Season to taste with more coarse kosher salt, if desired.
Melt butter in heavy small skillet over medium-high heat. Add pinch of coarse kosher salt to skillet and cook until butter browns, swirling skillet frequently to keep butter solids from burning, about 3 minutes.
Remove skillet from heat and stir in fresh lemon juice.
Transfer roasted radishes to warmed shallow serving bowl and drizzle brown butter over. Sprinkle with chopped radish tops and serve.